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OSMANIA UNIVERSITY " ^ No. Accession '
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I'*
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Title
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This book should be returned on or before the date las^
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bclo*v.
ELEMENTS OF ALGEBRA
THE MACM1LLAN COMPANY NKVV YORK
-
PAI-I.AS
BOSTON CHICAGO SAN FRANCISCO
MACMILLAN & CO, LONDON
LIMITKU HOMBAY CALCUTTA MELUCK'KNK
THE MACMILLAN
CO. OF TORONTO
CANADA,
LTD.
ELEMENTS OF ALGEBRA
BY
ARTHUR
SCJBULIi/TZE,
PH.D.
FORMERLY ASSISTANT PROFESSOR OF MATHEMATICS, NKW YORK ITNIVEKSITT HEAD OF THK MATHEMATICAL DKI'A KTM EN T, HIH SCHOOL OF COMMERCE, NEW 1 ORK CUT
THE MACMILLAN COMPANY 1917 All rights reserved
COPYRIGHT,
BY
1910,
THE MACMILLAN COMPANY.
Set up and electrotyped.
Published
May,
1910.
Reprinted
February, January, 1911; July, IQJS January, 1915; May, September, 1916; August, 1917.
September, 1910
.
;
;
Berwick & Smith Co. Norwood, Mass., U.S.A.
J. 8. Cushlng Co.
1913,'
PREFACE IN
this
book the attempt
in algebra,
with
all
while
still
made
to shorten the usual course
giving to the student complete familiarity
the essentials of the subject.
similar to the author's to its peculiar aim,
"
While
in
Elementary Algebra,"
many
respects
this book,
has certain distinctive features, chief
which are the following 1.
is
owing
among
:
All unnecessary methods
and "cases" are
omitted.
These
omissions serve not only practical but distinctly pedagogic " cases " ends. Until recently the tendency was to multiply as far as possible, in order to make every example a
social
case of a memorized method.
Such a large number of methods,
however, not only taxes a student's memory unduly but in variably leads to mechanical modes of study. The entire study of algebra becomes a mechanical application of memorized rules,
while the cultivation of the student's reasoning power is neglected. Typical in this respect is the
and ingenuity
treatment of factoring in
methods which are of
many
text-books
In this book
all
and which are applied in advanced work are given, but "cases" that are taught only on account of tradition, short-cuts that solve only examples real value,
manufactured for this purpose, etc., are omitted. All parts of the theory whicJi are beyond the comprehension
specially 2.
of
the student or wliicli are logically
practical
teachers
know how few
unsound are
omitted.
All
students understand and
appreciate the more difficult parts of the theory, and conse-
PREFACE
vi
quently hardly ever emphasize the theoretical aspect of alge bra. Moreover, a great deal of the theory offered in the averis logically unsound ; e.g. all proofs for the sign text-book age
two negative numbers, all elementary proofs theorem for fractional exponents, etc.
of the product of of the binomial 3.
TJie exercises are slightly simpler than in the larger look.
The best way to introduce a beginner to a new topic is to offer Lim a large number of simple exercises. For the more ambitious student, however, there has been placed at the end of the book a collection of exercises which contains an abundance
of
more
difficult
cises in this
work.
book
With very few
differ
bra"; hence either book 4.
from those
may
exceptions
in the
all
the exer
"Elementary Alge-
be used to supplement the other.
Topics of practical importance, as quadratic equations and
graphs, are placed early in the course.
enable students
This arrangement will of time to
who can devote only a minimum
algebra to study those subjects which are of such importance for further work.
In regard
may
to
some other features of the book, the following
be quoted from the author's "Elementary Algebra":
"Particular care has been bestowed upon those chapters in the customary courses offer the greatest difficulties to
which
the beginner, especially problems and factoring. The presenwill be found to be tation of problems as given in Chapter
V
quite a departure from the customary way of treating the subject, and it is hoped that this treatment will materially diminish the difficulty of this topic for young students. " The book is designed to meet the requirements for admis-
sion to our best universities
and
colleges, in particular the
requirements of the College Entrance Examination Board. This made it necessary to introduce the theory of proportions
PREFACE
vii
and graphical methods into the first year's work, an innovation which seems to mark a distinct gain from the pedagogical point of view.
"
By studying proportions during the first year's work, the student will be able to utilize this knowledge where it is most needed,
viz. in
geometry
;
while in the usual course proportions
are studied a long time after their principal application. " Graphical methods have not only a great practical value,
but they unquestionably furnish a very good antidote against 'the tendency of school algebra to degenerate into a mechanical application of
memorized
rules.'
This topic has been pre-
sented in a simple, elementary way, and of the
modes of representation given
it is
hoped that some
will be considered im-
provements upon the prevailing methods. The entire work in graphical methods has been so arranged that teachers who wish a shorter course
may omit
these chapters."
Applications taken from geometry, physics, and commercial are numerous, but the true study of algebra has not been sacrificed in order to make an impressive display of sham life
applications. to solve a
It is
undoubtedly more interesting for a student
problem that results in the height of Mt.
McKinley
than one that gives him the number of Henry's marbles. But on the other hand very few of such applied examples are genuine applications of algebra,
nobody would find the length Etna by such a method,
of the Mississippi or the height of Mt.
and they usually involve difficult numerical calculations. Moreover, such examples, based upon statistical abstracts, are frequently arranged in sets that are algebraically uniform, and hence the student is more easily led to do the work by rote
than when the arrangement braic aspect of the problem.
is
based principally upon the alge-
PREFACE
viii
It is true that
problems relating to physics often
offer
a field
The average
pupil's knowlso small that an extensive use of
for genuine applications of algebra.
edge of physics, however, is such problems involves as a rule the teaching of physics by the teacher of algebra.
Hence the
field of
genuine applications of elementary algebra work seems to have certain limi-
suitable for secondary school tations,
give as
but within these limits the author has attempted to
many
The author
simple applied examples as possible. desires to acknowledge his indebtedness to Mr.
William P. Manguse for the careful reading of the proofs and for
many
NEW
valuable suggestions.
YORK,
April, 1910.
ARTHUR SCHULTZE.
CONTENTS CHAPTER
I
PAGB
INTRODUCTION
1
Algebraic Solution of Problems Negative Numbers
1
3
Numbers represented by Letters Factors, Powers, and Hoots
....... ...
Algebraic Expressions and Numerical Substitutions
CHAPTER
15
........ ....
Subtraction
III
...
MULTIPLICATION
Numbers
Monomials
Multiplication of a Polynomial by a
10
22
29
CHAPTER
Multiplication of
15
27
Signs of Aggregation Exercises in Algebraic Expression
Multiplication of Algebraic
7
10
II
ADDITION, SUBTRACTION, AND PARENTHESES Addition of Monomials Addition of Polynomials
6
Monomial
31 31
....
34
35
Multiplication of Polynomials
36
Special Cases in Multiplication
39
CHAPTER IV 46 46
DIVISION Division of Monomials
Division of a Polynomial by a Monomial Division of a Polynomial by a Polynomial Special Cases in Division ix
47
48 61
X
CONTENTS CHAPTER V PAGE
,63
LINEAR EQUATIONS AND PROBLEMS
.....,.
Solution of Linear Equations
Symbolical Expressions
Problems leading
to
55 67
63
Simple Equations
CHAPTER VI FACTORING
Type
76 I.
Type II. Type III. Type IV. Type V. Type VI.
Summary
Polynomials, All of whose Terms contain a mon Factor
Quadratic Trinomials of the Quadratic Trinomials of the
Com77
Form x'2 -f px -f q Form px 2 -f qx + r
The Square of a Binomial x 2 Ixy The Difference of Two Squares Grouping Terms
.
.
.... -f
/^
.
.
.
78
80 83
84 86 87
of Factoring
CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
.
.
Common Factor Lowest Common Multiple
CHAPTER
89
89
Highest
91
VIII 93
FRACTIONS Reduction of Fractions Addition and Subtraction of Fractions
93 97
102
Multiplication of Fractions Division of Fractions
104
Complex Fractions
*
,
*
.
105
CHAPTER IX FRACTIONAL AND LITERAL EQUATIONS
......
112
Literal Equations
Problems leading to Fractional and Literal Equations
108 108
Fractional Equations .
.114
CONTENTS
XI
CHAPTER X
RATIO AND PROPORTION
.........
PAGE
120
Ratio
120
Proportion
121
CHAPTER XI SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE Elimination by Addition or Subtraction Elimination by Substitution Literal Simultaneous Equations Simultaneous Equations involving More than
....
129 130 133 138
Two Unknown
....
140
....
148
Graphic Solution of Equations involving One Unknown Quantity Graphic Solution of Equations involving Two Unknown Quan-
168
Quantities
Problems leading to Simultaneous Equations
CHAPTER
143
XII
GRAPHIC REPRESENTATION OF FUNCTIONS AND EQUATIONS Representation of Functions of One Variable
.
.
164
160
tities
CHAPTER
XIII
INVOLUTION
165
Involution of Monomials
165
Involution of Binomials
166
EVOLUTION
...
CHAPTER XIV 169
Evolution of Monomials
170
.
Evolution of Polynomials and Arithmetical Numbers
.
.
171
.
1*78
CHAPTER XV QUADRATIC EQUATIONS INVOLVING ONB UNKNOWN QUANTITY Pure Quadratic Equations
178
Complete Quadratic Equations Problems involving Quadratics
181
Equations in the Quadratic Character of the Roots
Form
189 191
193
CONTENTS
xii
CHAPTER XVI PAGK 195
THE THEORT OP EXPONENTS Fractional and Negative Exponents Use of Negative and Fractional Exponents
....
195
200
CHAPTER XVII RADICALS
205
206
Transformation of Radicals Addition and Subtraction of Radicals
210
.212
Multiplication of Radicals Division of Radicals
.....
Involution and Evolution of Radicals
219
Square Roots of Quadratic Surds Radical Equations
CHAPTER
214
218 221
XVIII
THE FACTOR THEOREM
227
CHAPTER XIX SIMULTANEOUS QUADRATIC EQUATIONS I.
II.
......
Equations solved by finding x +/ and x / One Equation Linear, the Other Quadratic
III.
Homogeneous Equations
IV.
Special Devices
232
.
.
.
232
.
.
.
234
236 237
Interpretation of Negative Results
and the Forms
i
-,
.
.
241
243
Problems
CHAPTER XX PROGRESSIONS
246
.
Arithmetic Progression Geometric Progression Infinite
24(j
251
263
Geometric Progression
CHAPTER XXI BINOMIAL THEOREM
.
BEVIEW EXERCISE
.
.
.
.
.
.
..
.
.
255
268
ELEMENTS OF ALGEBRA
ELEMENTS OF ALGEBRA CHAPTER
I
INTRODUCTION 1.
Algebra
may
it
arithmetic,
be called an extension of arithmetic. Like numbers, but these numbers are fre-
treats of
quently denoted by problem.
letters,
as illustrated in
the following
ALGEBRAIC SOLUTION OF PROBLEMS 2.
Problem.
is five
The sum
two numbers is 42, and the greater Find the numbers. the smaller number. of
times the smaller. '
x
Let
5 x = the greater number, 6x the sum of the two numbers.
Then and
6x
Therefore,
= 42,
x = 7, the smaller number, 5 x = 35, the greater number.
and 3.
A problem
4.
An
is
a question proposed for solution.
equation is a statement expressing the equality of
quantities; as,
6 a?
two
= 42.
In algebra, problems are frequently solved by denoting numbers by letters and by expressing the problem in the form of an equation. 5.
6.
Unknown numbers
letters of the alphabet
are employed. B
;
are usually represented as, x, y,
1
z,
by the
last
but sometimes other letters
ELEMENTS OF ALGEBRA
2
EXERCISE
1
Solve algebraically the following problems 1.
The sum
numbers is 40, and the greater Find the numbers.
of two
times the smaller.
A man
:
is
four
and a carriage for $ 480, receiving for the horse as for the carriage. much did he receive for the carriage ? 2.
twice as
3.
A
sold a horse
How
much
and
B own
vested twice as
a house worth $ 14,100, and
much
capital as B.
How much
A
has
in-
has each
invested ? 4.
The population
of
South America
is
9 times that of
Australia, and both continents together have 50,000,000 inFind the population of each. habitants.
The
and fall of the tides in Seattle is twice that in their sum is 18 feet. Find the rise and fall and Philadelphia, 5.
rise
of the tides in Philadelphia. 6.
Divide $ 240 among A, B, and C so that A may receive much as C. and B 8 times as much as C.
6 times as
A pole 56 feet high was broken so that the part broken was 6 times the length of the part left standing. .Find the length of the two parts. 7.
off
8.
If
The sum
two
of the sides of a triangle equals 40 inches. sides of the triangle are equal, and each is twice the A remaining side, how long is each side ?
A
9.
The sum
triangle is are equal,
of the three angles of any 180. If 2 angles of a triangle and the remaining angle is 4
times their sum,
how many
degrees are
there in each ?
B
G 10. The number of negroes in Africa 10 times the number of Indians in America, and the sum of both is 165,000,000. How many are there of each ?
is
INTRODUCTION
3
Divide $280 among A, B, and C, so that much as A, and C twice as much as B.
11.
B may
receive
twice as
Divide $90 among A, B, and C, so that B may receive much as A, and C as much as A and B together.
12.
twice as
A
13.
which
is
line 20 inches long is divided into two parts, one of long are the parts ? equal to 5 times the other.
How
A
travels twice as fast as B, and the tances traveled by the two is 57 miles. 14.
sum
of the dis-
How many
A, B, C, and
15.
does
A
much
take, if
B
and
D
as B,
miles did
4
each travel ?
D buy $ 2100 worth of goods. How much buys twice as much as A, C three times as
six times as
much
NEGATIVE NUMBE EXERCISE
2
Subtract 9 from 16.
1.
2.
Can 9 be subtracted from 7 ?
3.
In arithmetic
4.
The temperature
What
is
why
cannot 9 be subtracted from 7 ? "*
\
noon is 16 ami at 4 P.M. it is 9 the temperature at 4 P.M.? State this as an at
of subtraction. 5.
less. 6.
The temperature
8.
4 P.M.
is
7, and
at 10 P.M.
it is
10
expressing the last
below zero) ? What then is 7 -10?
answer 7.
at
What is the temperature at 10 P.M. ? Do you know of any other way of (3
Can you think
of
any other
practical examples
require the subtraction of a greater
which
number from a smaller
one? 7.
Many
greater
practical examples require the subtraction of a one, and in order to express in
number from a smaller
a convenient form the results of these, and similar examples,
ELEMENTS OF ALGEBRA
4
it becomes necessary to enlarge our concept of number, so as to include numbers less than zero.
8. Negative numbers are numbers smaller than zero; they are denoted by a prefixed minus sign as 5 (read " minus 5 "). Numbers greater than zero, for the sake of distinction, are fre;
quently called positive numbers, and are written either with a prefixed plus sign, or without any prefixed sign as -f- 5 or 5. ;
The
fact that a
below zero
thermometer falling 10 from 7 indicates 3
may now
be expressed 7 -10
= -3.
Instead of saying a gain of $ 30, and a loss of $ 90 we may write
is
equal to a
loss of $ 60,
$30 The
9.
-$90 = -$60.
6,
It is convenient for
10.
number
absolute value of a
without regard to its sign. 5 is The absolute value of
is
the number taken
of -f 3 is 3.
many
discussions to represent the
numbers by a succession of equal distances laid off on from a point 0, and the negative numbers by a similar
positive
a line
series in the opposite direction. ,
I
-6
I
-5
lit -4
-2
-3
I
I
I
+\
+2
I
-1
Thus, in the annexed diagram, the line from the line from
to
4,
I
I
+4
4-5
y
I
+6
to 4- 6 represents 4- 5,
1
etc.
left.
equals 4, 5 subtracted from
EXERCISE 1.
3
The addition of 3 is repspaces toward the right, and the subtrac-
4 represents
resented by a motion of "three tion of 8 by a similar motion toward the
Thus, 5 added to
I
+
If in financial transactions
we
1 equals
6, etc.
3
indicate a man's income
by
a positive sign, what does a negative sign indicate ? 2. State in what manner the positive and negative signs may be used to indicate north and south latitude, east and west
longitude, motion upstream
and downstream.
INTRODUCTION 3.
If north latitude
is
indicated by a positive sign, by what
is
south latitude represented ?
is
north latitude represented
4.
If south latitude
5.
What
6.
A
is
5
indicated by a positive sign, by what ?
the meaning of the year 6 yards per second ? erly motion of is
20 A.D. ?
merchant gains $ 200, and loses $ 350. - 350. (b) Find 200
Of an
(a)
east-
What
is
his total gain or loss ? 7.
If the temperature at 4 A.M. is 8 and at 9 A.M. it is 7 what is the temperature at 9 A.M. ? What, therefore,
higher, is 8
- +7? 8. A vessel
sails
journey. 9.
sails
A 22
(6)
11. 12. 13.
14. 15.
16. 17.
26.
from a point in 25 north latitude, and Find the latitude at the end of the
(a)
Find 25 -38.
vessel starts from a point in 15 south latitude, and due south, (a) Find the latitude at the end of the
journey, 10.
starts
38 due south,
(b)
Subtract 22 from
From 30 subtract 40. From 4 subtract 7. From 7 subtract 9. From 19 subtract 34. From subtract 14. From 12 subtract 20. 2 subtract 5. From 1 subtract 1. From
15.
24.
To 6 2 To To 1 From 1 To - 8 To 7 From
25.
Add
18.
19. 20.
21. 22.
23.
add
1.
add
2.
subtract 2.
add
9.
add
4.
1 subtract 2.
1 and 2.
Solve examples 16-25 by using a diagram similar to 10, and considering additions and subtractions as
the one of
motions.
(a)
Which is the greater number lor -1? (b) -2 or -4?
28.
By how much
27.
12.
add
is
:
7 greater than
12 ?
ELEMENTS OF ALGEBRA 29.
Determine from the following table the range of tempera-
ture in each locality
:
NUMBERS REPRESENTED BY LETTERS 11. For many purposes of arithmetic it is advantageous to express numbers by letters. One advantage was shown in 2 others will appear in later chapters ( 30). ;
EXERCISE 1.
2. 3.
and
b
4. 5.
many
If the letter
=
What
the value of
is
the value of 17
c,
= 5?
if c
ifc
5t?
if
a=
6,
= -2?
boy has 9c? marbles and wins 4c marbles has. he ? If a
Is the last
A
marbles,
answer correct for any value of d ? m dollars and lost 11 m
merchant had 20 much has he left ?
8.
What
9.
Find the numerical value
10.
is
4?
6.
that
4
means 1000, what
What is the value of 3 6, if b = 3 ? if b = 4 ? What is the value of a + &, if a = 5, and 6 = 7?
7.
How
t
is
the
sum
of 8 &
If c represents a certain
number ?
and G
how
dollars.
b ?
of the last
answer
if b
= 15.
number, what represents 9 times
INTRODUCTION
if
11.
From 26 w
12.
What is the numerical
1
subtract 19 m.
value of the last answer
if
m = 2?
m = -2? 13.
From 22m
of the answer
if
subtract
m=
1
25m, and
14.
Add
15.
From
16.
Add -lOgand +20 q. From 22# subtract 0.
19.
find the numerical value
2.
13 p, 3p, 6p, and subtract 24 p from the sum.
10 q subtract 20
17.
q.
18.
From subtract 26 Add - 6 x and 8 x.
From
20.
x.
Wp subtract 10^).
What sign, therefore, 140. 21. If a = 20, then understood between 7 and a in the expression 7 a ? 7 a=
is
FACTORS, POWERS, AND ROOTS The
12.
signs of addition, subtraction, multiplication, division, in algebra as they have
and equality have the same meaning in arithmetic. 13.
If there is no sign between
number, a sign of multiplication 6
x a
is
generally written 6 a
Between two (either
x
or
14.
Since 24
=
Similarly,
15.
thus,
A
x n
a letter and a
is
written win.
however, a sign of multiplication has to be employed as, 4x7, or 4 7. ;
written 47, for 47
A product is
two or more
m
letters, or
understood.
figures,
)
4x7 cannot be
;
two
is
means 40
-f 7.
the result obtained by multiplying together
quantities, each of which is a factor of the product. 3 x 8, or 12 x 2, each of these numbers is a factor of 24.
7, a, 6,
is
6 aaaaaa, or a ,
c are factors of 7 abc.
is the product of two or more equal factors called the " 5th power of a," and written a5 " the 6th is power of a," or a 6th.
power
aaaaa
and
;
;
The second power is also called the square, and the third 2 power the cube; thus, 12 (read "12 square") equals 144.
ELEMENTS OF ALQEBEA
8 16.
The
base of a
is
power
number which
the
is
repeated
as a factor.
The base
of a 3
is a.
17. An exponent is the number which indicates how many times a base is to be used as a factor. It is placed a little above and to the right of the base.
The exponent
of
m
6
is
6
n
;
is
the exponent of an
EXERCISE 1.
Write and
2.
72
.
5
find the numerical value of the square of 7, the cube of 6, the fourth power of 3, and the fifth power of 2. Find the numerical values of the following powers :
If
6.
.
42
10.
.
11.
3.
2*.
7.
2*.
4.
52
.
8.
10 6
5.
83
.
9.
I 30
a=3, 6=2, c=l, and 3
10
18.
ci
.
20.
c
19.
b2
.
21.
d\
28.
If
29. 30.
.
.
.
d=^ 22.
23.
O
.
9 .
2
12.
(4|)
13.
(1.5)
.
2 .
14.
25 1
15.
.0001 2
16.
l.l 1
17.
22
.
.
.
+3
2 .
find the numerical values of:
a*. 2
(6cf)
.
3
24.
(2 c)
25.
ab.
.
26.
2
27.
(4 bdf.
at).
= 8, what is the value of a? If m = what is the value of m ? = If 4 64, what is the value of a ? a3
2
-jJg-,
In a product any factor product of the other factors. 18.
In 12 win 8/), 12 19.
8
(i)
A
is
the coefficient of
numerical coefficient
is
is
called the coefficient of the
mw 8p,
12
m is the coefficient of n*p.
a coefficient expressed entirely
in figures. In
17
When
aryx,
17
is
the numerical coefficient.
a product contains no numerical coefficient, 1 1 a, a Bb 1 a*b.
stood ; thus a
=
=
is
under-
INTRODUCTION
9
20. When several powers are multiplied, the beginner should remember that every exponent refers only to the number near which it is placed. 2
3
means 3
aa, while (3
2
)
=3ax
3 a.
= 9 abyyy. 2* xyW = 2-2.2.2. xyyyzz.
9
afty
3
1 abc*
7 abccc.
EXERCISES If
a
= 4, b = 1, c = 2, and x = ^, find the
numerical values of
:
A
21. root is one of the equal factors of a power. According to the number of equal factors, it is called a square root, a cube root, a fourth root, etc. 3
is
6
is
the square root of 9, for 32 = 9. the cube root of 125, for 6 8 = 125.
a
is
the
root of a 5 the nth root of a".
fifth
,
indicated by the symbol >/""; thus Va is the is the cube root of 27, \/a, or more simply the square root of a.
The nth
root
fifth root of a,
Va,
is
Using
(Va) 22.
n
this
= a. The
is
A/27
symbol we
index of a root
root is to be taken. sign. In v/a, 7 23.
is
The
bracket,
[ ]
may is
express the definition of root by the
number which
what
the index of the root.
signs of aggregation are ;
indicates
It is written in the opening of the radical
the brace,
j
j
;
:
the parenthesis,
and the vinculum,
.
( )
;
the
ELEMENTS OF ALGEBRA
10
They are used, as in arithmetic, to indicate that the expres* sions included are to be treated as a whole. Each 10
is
b) is
(a
1],
sometimes read "quantity a
EXERCISE
= 2, b = 3, c = 1, d
If a
+
x (4 -f 1), 10 x [4 by 4 + 1 or by 5.
of the forms 10
to be multiplied
0,
x
10 x
4"+T indicates
that
b."
7
9, find the numerical value of:
1.
Vff.
7.
Val
13.
4(a
+ &).
2.
V36".
8.
-\fi?.
14.
6(6
+ c).
3.
V2a.
9.
4V3~6c.
15.
(c-f-d)
4.
v'Ta.
10.
5Vl6c.
16.
6.
\/c.
11.
aVc^.
17.
6.
V^a6.
12.
2
[6-c]
.
3 .
AND NUMERICAL
ALGP:BRAIC EXPRESSIONS
SUBSTITUTIONS
An
24.
algebraic expression is a collection of algebraic
bols representing
A
25.
some number
monomial or term
separated by a sign (6
+ c + d}
26.
is
or
is
e.g.
;
6 a26
7
Vac
2
an expression whose parts are not as 3 cue2,
9
~* Vx,
o c ^and a monomial, since the parts are a (6 + -f-
A polynomial is an
;
c -f d).
expression containing more than one
term. y,
27.
a2
+
28.
A binomial is 62 ,
3
!^-f\/0-3
and |
-
ft,
and a 4
+ M -f c
4 -f-
d 4 are polynomials.
a polynomial of two terms.
\/a are binomials.
A trinomial
is
a polynomial of three terms.
V3
sym-
-f 9.
are trinomials.
INTRODUCTION In a polynomial each term
29.
is
11
treated as
were con-
if it
tained in a parenthesis, i.e. each term has to be computed before the different terms are added and subtracted. Otherwise operations of addition, subtraction, multiplication, and division are to be performed in the order in which they are written all
from
left to right.
E.g. 3
Ex.
4
_|_
.
5
means 3
4-
20 or 23.
28
Find the value of 4
1.
+5
32
-
*^.
= 32 + 45-27 = 50. Ex.
If a = 5, b = 3, c = 2, d = 0, - 9 aWc + f a b - 19 a 6cd
2.
2 of 6 ab
3
find the numerical value
2
6 aft 2 - 9 a& 2 c + f a 6 - 19 a 2 bcd = 6 5 32 - 9 5 32 2 + ^ 5 8 3 - 19 = 6. 5- 9-9. 6- 9- 2 + I-126- 3-0 = 270 - 810 + 150 = - 390. EXERCISE 8* 3
-
If
.
a=4, 5=3, c=l, d=Q, x=^,
2.
+ 26+3 c. 3a + 56
3.
a 2 -6.
4.
a2
5.
5a2
6.
2 a2
7.
6a2 +4a62 ~6c'
8.
27
1.
a
.
52
3
.
2
find the numerical value of: 9.
2
.
5c6 2 +-6ac3 3
8
3
17c3
-d
a
11.
3a& 2 + 3a2 6-a&c2
-f & -f c
-hl2o;.
s
10.
.
'
-5c
c
2
+-d
2
12.
.
-46c-f2^^ + 3 a& +- 4 6^9 ad. 3
- 5 ax
.
l
-+12a(i
50 a6cd.
4 .
13.
(a
14.
(a -f b)
*15. 16.
a2
+ (a + 6)c 6 (2 + a 2
c
2
-f
).
4a6-fVa-V2^.
* For additional examples see page 268,
2 .
ELEMENTS OF ALGEBRA
12 &
17
18
*
'
8
Find the numerical value of 8 a3
22.
a = 2, 6 = 1. a = 2, 6 = 2.
23.
a =3,
24.
a=3,
21.
25.
26. 27.
6=2.
28.
6 = 4. = = 5. a 3, 6
30.
29.
Express in algebraic symbols 31. Six times a plus 4 times 32. 33.
-f-
6s, if
6 a6 2
:
a = 3, 6 = 3. a = 4, 6 = 5. a =4, 6 = 6. a = 3, 6 = 6. a = 4, 6 = 7.
:
6.
Six times the square of a minus three times the cube of Eight x cube minus four x square plus y square.
w
cube plus three times the quantity a minus
34.
Six
35.
The quantity a
minus
12 cr6
6
plus 6 multiplied
6,
6,
2 by the quantity a
2 .
Twice a3 diminished by 5 times the square root of the quantity a minus 6 square. 36.
37.
Read the expressions
38.
What kind of expressions are Exs. 10-14
of Exs. 2-6 of the exercise. of this exercise?
The
representation of numbers by letters makes it posvery briefly and accurately some of the principles of arithmetic, geometry, physics, and other sciences. 30.
sible to state
Ex. a, 6,
If the three sides of a triangle contain respectively c feet (or other units of length), and the area of the
and
triangle
then
is
S
square feet (or squares of other units selected),
8 = \ V(a + 6 + c) (a 4- 6 - c) (a - 6 -f c) (6
a
+ c).
INTRODUCTION
15
13
E.g. the three sides of a triangle are respectively 13, 14, 15 therefore feet, then a 13, b 14, and c
=
=
=
and
;
S = | V(13-hl4-fl5)(13H-14-15)(T3-14-i-15)(14-13-f-15)
= V42-12-14.16 1
= 84,
i.e.
the area of the triangle equals
84 square
feet.
EXERCISE
9
The
distance s passed over by a body moving with the uniform velocity v in the time t is represented by the formula 1.
Find the distance passed over by A snail in 100 seconds, if v .16 centimeters per second. A train in 4 hours, if v = 30 miles per hour. b. c. An electric car in 40 seconds, if v = 50 meters per second 5000 feet per minute. d. A carrier pigeon in 10 minutes, if v :
a.
2. A body falling from a state of rest passes in t seconds 2 over a space S (This formula does not take into ac^gt 32 feet, count the resistance of the atmosphere.) Assuming g .
=
(a)
How
far does
a body fall from a state of rest in 2
*
seconds ?
A
stone dropped from the top of a tree reached the ground in 2-J- seconds. Find the height of the tree. How far does a body fall from a state of rest in T ^7 of a (c) (b)
second ? 3.
By
using the formula
find the area of a triangle
whose
(a) 3, (b) 5, (c) 4,
sides are respectively
4, and 5 feet. 12, and 13 inches. 13, and 15 feet.
ELEMENTS OF ALGEBRA
14 4.
If
meters,
the radius of a circle etc.),
the area
square meters,
etc.).
n
If
i
i
6.
2 inches.
(b)
=p
Find by means
(b)
It
represents the simple interest of
years, then
(a)
units of length (inches,
2
square units (square inches, Find the area of a circle whose radius is
(a) 10 meters. 5.
H
is
$ = 3.14
The The
n
interest on
p
dollars at r
fo
in
*
r
or
%>
of this formula
interest
5 miles.
(c)
$800
:
for 4 years at
ty%.
on $ 500 for 2 years at 4 %.
If the diameter of a sphere equals d units of length, the
$=
2
3.14d (square units). (The number 3.14 is frequently denoted by the Greek letter TT. This number cannot be expressed exactly, and the value given above is only an surface
approximation.) Find the surface of a sphere whose diameter equals (a) 7.
8000 miles. If the
(b) 1 inch.
diameter of a sphere equals d
volume
V=
~
:
10
(c)
feet,
feet.
then the
7n
cubic feet.
6
Find the volume of a sphere whose diameter equals: (a) 10 feet.
(b)
3
feet.
(c)
8000 miles.
F
denotes the number of degrees of temperature indi8. If cated on the Fahrenheit scale, the equivalent reading C on the Centigrade scale may be found by the formula y
C
= f(F-32).
Change the following readings (a)
122 F.
(b)
to Centigrade readings:
32 F.
(c)
5
F.
CHAPTER
II
ADDITION, SUBTRACTION, AND PARENTHESES
ADDITION OF MONOMIALS 31.
While
word sum
in arithmetic the
refers only to
the
result obtained
by adding positive numbers, in algebra this word includes also the results obtained by adding negative, or positive and negative numbers. In arithmetic we add a gain of $ 6 and a gain of $ 4, but we cannot add a gain of $0 and a loss of $4. In algebra, however, we call the aggregate value of a gain of 6 and a loss of 4 the sum of the two. Thus a gain of $ 2 is considered the sum of a gain of $ 6 and a loss of $ 4. Or in the symbols of algebra $4) = Similarly, the fact that a loss of loss of
$2 may be
+ $2.
$6 and a gain
of
$4
equals a
represented thus
In a corresponding manner we have for a loss of $6 and a of
$4
(- $6) + (-
$4) = (-
loss
$10).
Since similar operations with different units always produce analogous results, we define the sum of two numbers in such a way that these results become general, or that
and
(+6) + (+4) = + 16
10.
ELEMENTS OF ALGEBRA
16 32.
These considerations lead to the following principle
:
If two numbers have the same sign, add their absolute values if they have opposite signs, subtract their absolute values and ;
(always) prefix the sign of the greater. 33.
The average
of two
numbers
average of three numbers average of n numbers is the
is one half their sum, the one third their sum, and the sum of the numbers divided by n.
is
Thus, the average of 4 and 8
The average The average
of 2, 12,
(-17)
18.
15
19.
is 0.
3 J.
-
0, 10, is 2.
10
of:
Find the values 17.
is
of 2, '- 3, 4, 5,
EXERCISE Find the sum
4
of:
+ (-14).
+ (-9). + -12.
20.
l-f(-2).
21.
(_
22.
In Exs. 23-26, find the numerical values of a + b 23.
a
24.
a
= 2, = 5,
6 6
= 3, c = = 5, c =
4,
5,
d = 5. d = 0.
-f c-j-c?, if :
ADDITION, SUBTRACTION, a
25.
26. 27.
30.
31.
= -23, c=14, & = 15, c = 0, &
1?
d = l. d=
3.
What number must be added to 9 to give 12? What number must be added to 12 to give 9 ? What number must be added to 3 to give 6 ? C* What number must be added to 3 to give 6? **j Add 2 yards, 7 yards, and 3 yards. }/ Add 2 a, 7 a, and 3 a. \\ Add 2 a, 7 a, and 3 a. -'
28. 29.
= 22, = -13,
AND PARENTHESES
-
'
32. 33.
Find the average of the following 34.
3 and 25.
^
35.
5 and
- 13.
36.
12,
13,
39.
-8
'
and
37.
2, 3,
38.
- 3,
sets of
numbers:
- 7, and 4, - 4, - 5,
13. 6,
- 7,
and
1.
4
F.,
2.
Find the average of the following temperatures 27 F., and 3 F.
:
F.,
40. Find the average temperature of New York by taking the average of the following monthly averages 30, 32, 37, :
48, 60, 09, 74, 72, 66, 55, 43, 34. 41. Find the average gain per year of a merchant, if his yearly gain or loss during 6 years was $ 5000 gain, $3000 gain, $1000 loss, $7000 gain, $500 loss, and $4500 gain. :
Find the average temperature of Irkutsk by taking the average of the following monthly temperatures 12, 10, -4, 1, 6, 10, 12, 10, 6, 0, - 5, -11 (Centigrade). 42.
:
34.
Similar or like terms are terms which have the same
literal factors, affected
6 ax^y and
7 ax'2 y, or
by the same exponents. 5 a2 &
and
,
or 16
Va + b
and
2Vo"+~&,
are similar terms.
Dissimilar or unlike terms are terms 4 a2 6c and o
4 a2 6c2 are dissimilar terms.
which are not
similar.
ELEMENTS OF ALGEBRA
18 35.
The sum
The sum
of 3
of
two similar terms
x 2 and
x2
is
f
another similar term.
is
x2 .
Dissimilar terms cannot be united into a single term. The indicated by connecting
sum of two such terms can only be them with the -f- sign. The sum The sum
of a of a
and a 2 and
is
a
b
is
-f-
a2
a
-f (
.
6), or
a
6.
In algebra the word sum is used in a 36. Algebraic sum. b wider sense than in arithmetic. While in arithmetic a denotes a difference only, in algebra it may be considered b. either the difference of a and b or the sum of a and The sum
of
2 a&, and 4 ac2
a,
a
is
EXERCISE
2 a&
-|-
4 ac2.
11
Add: 1.
-2 a +3a -4o
2.
ab
7
xY xY 7 #y
12
6.
7.
Find the sum of 9.
\
-f-
2 ,
-f
4 a2,
5 a2
2 wp2 - 13 rap
12
10.
dn
7 a 2 frc
:
-3a
2 a2,
2
1
13 b sx
c
,
11.
2(a-f &),
12.
5l
13.
Vm
-f- ii,
,
3(a-f-6),
5Vm + w,
,
+ 6 af
.
25 rap 2, 7 rap2. 9(a-f-6),
12Vm-f-n,
12(a-f b)
14
AND PARENTHESES
ADDITION, SUBTRACTION, Simplify
19
:
15.
-17c + 15c8 + 18c + 22c3 +c3
17.
3
xyz
3
+ xyz
12 xyz
.
+ 13 xyz + 15 xyz.
Add: 18.
ra
19.
+m """
ZL
n
n
2
21.
2
a a8
x*
**,
22.
m
20.
6
23.
c
^24.
2
^
25.
7
2
-1
i
1
-co*
l^S
26.
mn
27.
xyz
mri
Simplify the following by uniting like terms: 29. 30.
3a-76 + 5a + 2a-36-10a+116. 2a -4a-4 + 6a -7a -9a-2a + 8. 2
2
2
31.
32. 33.
"Vx + y
Vaj + y 2
2 Vi
+ + 2 Va; + / + 3 Va; + y. 2/
Add, without finding the value of each term 34.
5x173 + 6x173-3x173-7x173.
35.
4x9'
-36.
:
10x38 ADDITION OF POLYNOMIALS
Polynomials are added by uniting their like terms. It convenient to arrange the expressions so that like terms may be in the same vertical column, and to add each column. 37.
is
ELEMENTS OF ALGEBRA
20
2 Thus, to add 26 ab - 8 abc - 15 6c, - 12 a& 4- 15 abc - 20 c 5 ab 4- 10 6c 6 c 2 and 7 a&c 4- 4 6c + c 2 we proceed as ,
,
f 110WS:
,
& c~15&c
26 aft- 8
-20c2
-12a&4l5a&c
- 6a& a5c
7
+
Numerical substitution
and
a, 5,
- 3 a -f 4
sum
convenient method for
offers a
To check
the addition of
c
any convenient
assign
ft
-f 5
c, e.g.
c
-f-
= 10
3, therefore the answer
correct.
is
NOTE. While the check is almost certain an absolute test e.g. the erroneous answer ;
equal
Sum.
2
c'
a = 1, 6=2, c = 1, = - 3 + 8 + 5 = 1 0, 2 0-25- c= 2- 4-1 = -3, 4 = 7. a 4- 2 6 + 4 c = 1 +4
numerical values to
then
But 7
ca
26
6c
checking the sum of an addition. 3 a -f 4 1) 4- o c and 4- 2 a 26
the
4
4 be
9a& 38.
6ca
-f-lO&c
to
show any
a406
It is
error,
4c would
not also
7.
In various operations with polynomials containing terms with different powers of the same letter, it is convenient to arrange the terms according to ascending or descending powers 39.
of that letter. 7 4.
x
4 5 x"2 + 7 x* 4 5 -7a &+4a
6 a7
of x.
fi
5 4
is
6c
arranged according to ascending powers 4 7 a&
,
we
shall not, at this
a factor of a 2
6
factor is said to be prime, if it contains
factors (except itself
The prime
and unity)
;
factors of 10 a*b are 2, 5,
76
otherwise ,
a, a, 6.
it is
2 .
no other
composite.
FACTORING 106.
the process of separating an expression expression is factored if written in the
is
Factoring
77
An
into its factors.
form of a product. 2 4 x + 3) is factored if written (x' would not be factored if written x(x and not a product.
The factors
107.
The prime
of a
factors of 12
4)
form
+3,
It (a; 8) (s-1). for this result is a sum,
monomial can be obtained by inspection
&V
2
Since factoring
108.
in the
is
are 3, 2, 2,
01,
x, x,
y.
?/,
the inverse of multiplication,
it fol-
lows that every method of multiplication will produce a method of factoring. E.g. since (a + 6) (a 2 IP factored, or that a
=
= a - 62 + &)(a 2
6)
(a
,
it
that a 2
follows
- 62
can be
&).
Factoring examples may be checked by multiplication by numerical substitution.
109.
or
TYPE
I.
POLYNOMIALS ALL OF WHOSE TERMS CONTAIN A COMMON FACTOR
mx + my+ mz~m(x+y + z). Ex.
110.
The
1.
Factor G ofy 2
greatest factor
common
6
and the quotient But, dividend
- 9 x if + 12 xy\
to all terms
a% - 9 x2 y 8 + 12
2 x2
is
2
3 xy
-f
55.)
4
flcy*
8
by
2
xy'
3
.
Divide
xy\
2 1/
.
= divisor x quotient. - 9 x2^ + 12 sy* = 3 Z2/2 (2 #2 - 3 sy + 4 y8).
6 aty 2
Hence
Ex.
is
(
2
2.
Factor
14 a*
W-
21 a 2 6 4 c2
+ 7 a2 6
2
c2
7
a2 6 2 c 2 (2 a 2
- 3 6a + 1).
ELEMENTS OF ALGEBRA
78
EXERCISE Resolve into prime factors
- 12 cdx.
1.
6 abx
2.
3x*-6x*.
3.
15
2
4
&-{-20a
4.
14a
5.
Ilro8
6
2
&3
39
:
6.
4 tfy -f- 5 x*y 2
7.
17 a? - 51 x4
8.
.
.
s
.
s
.
2
2
2
.
9. -7a & 10. + llm -llm. 11. 32 a *?/ - 16 a'V -f 48 ctfa^ 2
2
4
6 xy
+ 34 X 8 a*b -f 8 6V - 8 c a 15 ofyV - 45 afy - 30 aty. a -a '-J-a 4
3
2
:
4
.
8 .
12.
13.
34
14.
a^c 8 - 51
aW + 68
a6c.
15.
16. 17. 18.
19.
q*-q*-q
2
+ q.
a(m-f-7i) + & ( m + 3 (a + 6) -3 /(a + 6). 7i
)-
2
a;
+ 13 -8.
21.
13- 5
22
2.3.4.5 + 2.3.4.6.
-
23.
2
3
5-f 2
.
3
5
6.
20.
TYPE
IT.
QUADRATIC TRINOMIALS OF THE FORM
111. In multiplying two binomials containing a common 3 and 5 to obterm, e.g. (as 3) and (cc-f-5), we had to add tain the coefficient of x, and to multiply 3 and 5 to obtain the term which does not contain x or (x 3)(x -f 5) 15. x2 -f-2 x
=
In factoring x2
15 we have, obviously, to find two numbers whose product is 15 and whose sum is -f- 2. 2 Or, in general, in factoring a trinomial of the form x -f-/>#-f q,
we have
to find
whose product
2x
-f
two numbers m and n whose sum is p and and if such numbers can be found, the y
is g;
factored expression
is
(x -}-m)(x
+ n).
FACTORING Ex.
Factor a2
l.
-4 x - 11.
We may consider or
77
1,
79
77 as the product of 1 77, or 7 11, or 11 and 7 have a sum equal to 4. .
11
7,
but of these only
Hence
a:
2
- 4 x - 77 =
(a;- 11) (a
+
7).
Since a number can be represented in an infinite number of ways as the sum of two numbers, but only in a limited number of ways as a product of two numbers, it is advisable to consider the factors of q first. If q is positive, the two numbers have both the same sign as p. If q is negative, the two numbers
have opposite
signs,
and the greater one has the same sign
as p. of this type, however, can be factored.
Not every trinomial Ex.
2.
Factor a2
- 11 a + 30.
The two numbers whose product and -6. a2
Therefore Check.
Ex.
If
3.
tf
30 and whose
sum
11 are
5
a 4- 30 = (a - 5) (a 6). + 30 = 20, and (a - 5) (a - G) = - 4 - 6 = 20. .
+ 10 ax - 11 a
2 .
11 a2 and whose sum The numbers whose product is and a. 2 11 a?=(x + 11 a) (a- a). Hence fc -f 10 ax
is
10 a are 11 a
12 /. Factor x? - 1 afy 8 The two numbers whose product is equal to 12 yp and whose sum equals 3 8 7 y are -4 y* and -3 y*. Hence z6 -? oty+12 if= (x -3 y)(x*-4 y ).
Ex.
-
is
11
a = 1, a 2 - 1 1 a
Factor
is
112.
+
4.
In solving any factoring example, the student should first all terms contain a common monomial factor.
determine whether
EXERCISE
40
Besolve into prime factors : 4.
tf-
5.
3.
m -5m + 6. 2
6.
a2 -
ELEMENTS OF ALGEBEA
80
x*-2x-8. + 2x-S.
22.
8.
x2
9.
y_ 6y
24.
7.
2
10.
?/
2
11.
?/
12.
?/
2
23.
16.
+6y
16.
-15?/
+
25.
44.
26.
-5?/-14.
27.
+ 4?/-21. + 30. or - 17 + 30. 2
13.
28.
?/
14.
15.
a 2 +11 a
29.
a?
30.
^
16.
2
2
a2
21.
a4
TYPE 113.
.
6
8
8
4
2
a;
x*y ra
-9a&-226 + 8 a -20.
.
2 .
2
ITT.
3
4
32.
2
.
2
4xy
21y. 21 a 2
4 wia 2
2
a' 2
.
- 70 x y - 180 2
34.
10 x y 2
35.
200 x2
36.
4 a 2 - 48
+
+ + 446
400 x aft
a;
2 .
200. 2 .
QUADRATIC TRINOMIALS OF THE FORM
According to 66, - 2) = 20 x2 + 7 x - 6. (4 x + 3) (5 x 20 x2 is the product of 4 a; and 5 x. 6 is the product of + 3 and 2. .
+7 Hence
.
2
4
33. 2
.
2
+ 7ax 18. -17a& + 7(U
r
_|_ ft)2
256
35. 2
38.
a5
39.
a
42 s 2
(a;
2
2 ?/)
.
4
V 2
51 xyz
+ 50.
(
1
40. 3 41.
.
__ G4.
a -128.
36. any
48. + 6 aft + 3 80 a 310 x 40. 4
20
34.
-50^ + 45. a3
32.
29.
8
4
tt
33. (^
.
n Qy 2 . 2
30.
27. 28.
.
-42 a + 9 a 20a -90a -50. 4
-2a + a*-l. 42 x - 85 xy + 42 y 10 w 43 w 9. 25 a + 25 aft - 24. 13 c - 13 c - 156.
26.
3#4 -3a2 -36.
CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
HIGHEST COMMON FACTOR 120.
The
common
highest
factor (IT. C. F.) of
two or more
the algebraic factor of highest degree common expressions to these expressions thus a 6 is the II. C. F. of a 7 and a e b 7 is
.
;
Two
expressions which have no are prime to one another. 121.
The H.
common
factor except unity
two or more monomials whose factors
C. F. of
are prime can be found by inspection. The H. C. F. of a 4 and a 2 b is a2 .
The H. The H.
C. F. of
aW, aW,
C. F. of (a
+
8 ft)
and
and (a
+
cfiW is 2
fc)
a 2 /) 2
(a
.
4 ft)
is
+ 6)
(a
2 .
122. If the expressions have numerical coefficients, find by arithmetic the greatest common factor of the coefficients, and prefix it as a coefficient to H. C. F. of the algebraic expressions. Thus the H. C. F. of 6 sfyz, 12 tfifz, and GO aty 8 is 6 aty.
The student should note H. C.
F. is the lowest
that the power of each factor in the power in which that factor occurs in any
of the given expressions.
EXERCISE Find the H. C. F. of
4.
2.
3.
15
aW,
13 aty
8 ,
25
48
:
W.
39 afyV.
5. 6.
89
33
3
-
,
5
2 7
22 3 2 2
5
3
s ,
7
24
23 3 ,
3 ,
2
2
2
5,
.
5 7
s
.
34 2s ,
.
II
2 .
54
-
32
.
ELEMENTS OF ALGEBEA
90
6 rarcV, 12 w*nw 8, 30
7.
mu\
39 afyV, 52 oryz4, 65 zfyV. 38 #y, 95 2/V, 57 a>V.
8. 9.
aWd,
10.
225
11.
9
4a
12.
4(m -f ?i)
10
8 a
,
a&X -15 bed
75
16 a
,
3 ,
11
24 a
,
5(w + w)
2 ,
8
6(m+l) (m+2),
14.
6
-
3 a;
(a7
5 ?/)
,
9
7(m + n}\m 2
8(?/i-f-l)
aj*(a?
.
6.
3
13.
2
- y)\
O+
12
0^(0;
3),
ri).
4(m+l)
- y)
2 .
3 .
123. To find the H. C. F. of polynomials, resolve each polynomial into prime factors, and apply the method of the preceding article. Ex. 1. Find the H. C. F. of + 4 if, x2
^-4^
and
tf
-7 xy + 10 f. - 3 xy + 2 y* = (x - 2 ?/) (x - y) - 7 xy + 10 2 = (x - 2 y) (a; - 5 y). = x 2 y.
x*
x2
Hence the H. C. F.
.
7/
EXERCISE Find theH. 1.
4 a3 6 4 8 a663 - 12 as 66
2.
15 x-y^ 2 10 arV - 5 x3?/ 2
3.
25 m27i, 15
4.
.
,
,
4
3ao;
49
C. F. of:
3
7/i
-3^
4
n2
10
mV.
6 mx - 6 4
,
.
4 ?io; .
5.
6 a2
6.
y?
7.
a2
8.
ar*
- 6 a&,
10.
12. 13. 14.
15. 16.
2
2
2
a
-
2
2
2
a;?/
,
2
2
,
a;
^-707 + 12, 0^-80:4-16, ^a + 5^ + 6,^-9, ^-f a;-6. a2 - 8 a + 16, a3 -16 a, a -3a-4. a2 + 2a-3, a2 + 7a-f!2, a3 -9a. y + 3y-64,y + y-42, 2a -f5a-f 2, 4a -f 4a2
-5^
- # 4 afy -f 4 - 6 a' + 2 a& + 6 - 5 + 6, ^-
9.
11.
5 a6
.
2 .
3 .
LOWEST COMMON MULTIPLE
91
LOWEST COMMON MULTIPLE
A
multiple of two or more expressions is an be divided by each of them without a which can expression 124.
common
remainder.
Common
2 multiples of 3 x
The
125.
lowest
and 6 y are 30 xz y, 60
common
2
x^y'
,
300 z 2 y,
etc.
two or more
multiple (L. C. M.) of
expressions is the common multiple of lowest degree; thus, ory is the L. C. M. of tfy and xy*. 126. If the expressions have a numerical coefficient, find by arithmetic their least common multiple and prefix it as a coefficient to the L. C.
The The
M of the algebraic expressions.
L. C.
M.
of 3
L. C.
M.
of 12(a
aW,
+
2
a^c8 3
ft)
,
6
c6 is
and (a
C a*b*c*.
+ &)*( -
&)
2
is
12(a
+ &)( - 6)2.
127. Obviously the power of each factor in the L. C. M. is equal to the highest power in which it occurs in any of the
given expressions. 128. To find the L. C. M. of several expressions which are not completely factored, resolve each expression into prime factors and apply the method for monomials.
Ex.
1.
Find the L.
C.
M.
of 4 a 2 6 2 and 4 a 4
4 a 2 &2
2.
.
_
Find the L.C.M. of as -&2 a2 + 2a&-f b\ and 6-a. ,
= (a -f
Hence the L.C.M. NOTE.
2
=4 a2 62 (a2 - 6 3 ).
Hence, L. C. M.
Ex.
-4 a 68
The
L. C.
M. of the
last
2
&)'
(a
-
6) .
- (a + &) 2 (a
In example ft). two lowest common multiples, which is
also
general, each set of expressions has
have the same absolute value, but opposite
signs.
ELEMENTS OF ALGEBRA
92
EXERCISE Find the L. 1. 2.
a,
a 2 a3 ,
C. .
xy\
afy, 3
y*. 2
4a
3.
2
7.
4 a 5 6cd, 20
8.
9
ic
a,
10.
,
,
50
M. of:
8 a.
3
5.
6
6.
afc'cd
3 ab, 3(a
4.
2
40 abJ, 8 d 5
,
+ b).
9.
6
-f
2
6 y, 5
5
a?
a2
2 a?-b\ a + 2ab + b' 2a-2b.
a~b,
b
a?
14.
.
2
7i)
,
3(m
18.
19.
x2
3
a,-
a;
-f2, x
2
a
5
4,
a;
-f-
2
3#
+ 2,
5
21.
a 2 -fa6, a&
5
#,
x2
3 a
20.
or -f-
+ 5 a + 6,
2
+&
22.
a -!, a^-1,
23.
ic
24.
ax -{-ay ~
7ic+10, bx
15
3, 2
a2
,
4
2
2 .
a 2 -f 4 a +4.
2
-f
x2 + 4 a
-f 4,
#.
~ab
6b 2
.
1. a?
2
8
lOaj-f-lfi, by,
3 a
a.
+ n) 4 m
-4)(a-2)
,
2
30
a, a.
2
+ 6. + 2, a -f 3, a 1. 2 a - 1, 4 a - 1, 4 a -f 2.
17.
G
,
,
?/.
2
15. 16.
2
2
ic
y,
a
b,
+
(a
,
13.
-{-
3 Z>
,
3 a
,
x.
.
,
a;
3
2(m
,
12.
a?b,
5 a 2 ^ 2 15
T a
(a-2)(a-3) ( a -3)(a-4) 2 2a?b-'2ab 2 a, 2 a
3
8 afy, 24
.
2
11.
xif,
3
b,
x*
~5a;-f 6.
2 x -\-2 y.
(For additional examples see page 268. )
2
CHAPTER
VIII
FRACTIONS
REDUCTION OF FRACTIONS
A
129.
with a
-f-
fraction is
an indicated quotient; thus -
is identical
The dividend a is called the numerator and the The numerator and the denominator
b.
divisor b the denominator.
are the terms of the fraction. All operations with fractions in algebra are identical
130.
with the corresponding operations in arithmetic. Thus, the value of a fraction is not altered by multiplying or dividing both its numerator and its denominator by the same number; the product of two fractions is the product of their numerators divided by the product of their denominators, etc. In arithmetic, however, only positive integral numerators shall assume that the
and denominators are considered, but we
arithmetic principles are generally true for
all
algebraic numbers.
131. If both terms of a fraction are multiplied or divided by the same number) the value of the fraction is not altered. rni
Thus
A
132.
and
Reduce
1.
Remove tor, as 8,
TT
= ma
b
mb
is
i
,
Hence
~-
successively all 2
a?,
j/' ,
and z 8
and
mx = -x my y
in its lowest
denominator have no
its
Ex.
fraction
a -
terms
common
when
its
numerator
factors.
to its lowest terms.
common
divisors of
(or divide the terms
6
2
.ry ^
24
2 z = --
3x
by
numerator and denomina-
their H. C. F.
ELEMENTS OF ALGEBRA
94
133. To reduce a fraction to its lowest terms, resolve numerator and denominator into their factors, and cancel all factors that are
common
Never cancel terms of the numerator or the
to both.
denominator; cancel factors only.
Ex.
2.
a*
Keduce
~
6 a'
6a qs
_. 6
tf
Ex.
3.
Keduce
a*
*8a
to its lowest terms.
24 a2
4
n2 + 8 a 24 a*
-
_ ap 2 - 6 a + 8) 6 d\a* - 4)
~ 2 62
--
to its lowest terms.
a2
62
_Q
2 6
EXERCISE 51* Reduce i
to lowest terms
9-5
:
3
o 3
*'
*
32
78
2.33
-7 a
2
'
12a4"
T^
3
3T5"**
'
36 arV 18 x2^'
* See page 268.
K 6
'
39 a2 6 8c4
FRACTIONS 7-
^-.
9.
10.
11
'
22 a 2 bc 1
8.
4-
m-
n h
g
95
2
m
3
11
21.
J-
~__ 9n _ 22
^+3*. LJZJ^JL.
9x +
23.
.
^ Mtr
"a"
04
.
!l
9
'-M
12.
3
3
??i
2
6
or
it*?/
2fi 25.
.
_
7i
-9 - 10 a + 3 2
rt c
a a2 c
-L
4.
&
c
a
_6
6
c
4-
ab 2 4-
4-
a6 2 4- &c 2
&c*
a ,
c
ac
a
a2c
~
2
4- afr 4- ^c
2
ELEMENTS OF ALGEBRA
16
147.
In
many examples
the easiest
mode
of simplification
ia
multiply both the numerator and the denominator of the mplex fraction by the L. C. M. of their denominators. If the numerator and denominator of the preceding examples multiplied by a&c, the answer is directly obtained.
B
x -}- ?/ x y _x^_l X ~V x+y .
Ex.
2.
Simplify
xy
x
.
+y
Multiplying the terms of the complex fraction by (x y), the expression becomes (x
EXERCISE Simplify
57
:
x 2.
y X
4* 2 y
3.
JL.
4.
+6.
&
6.
7i+~ .
,a
.
c
^c
-n a
7.
a
9.
_^
,y
.
m ""
-.
32
a
8.
i. c
x*
10.
FRACTIONS
m 11.
,
o
1 15.
:
~
*
a "~ 2
12.
1
i
~T"
!^-5n
107
y 1
-
-
&
a
a
+2
1
i
2
4
'
5
,
-i
19
20
-
a4-6 13.
m^n* n
17.
i
1
1
a
+
"
"
f
(
1 /*-_i_i
4-
14.
L
s-y
18.
1
+
1+ 1 ti
(For additional examples see page 273.)
flg-f-l
a?l
a;-~l
ic+1
CHAPTER IX FRACTIONAL AND LITERAL EQUATIONS
FRACTIONAL EQUATIONS If an equation contains fracbe removed by multiplying each term by the may L. C. M. of the denominator. 148.
Clearing of fractions.
tions, these
^-2^ 63 =
Ex.1. Solve
-* + *-*.
12
2
Multiplying each term by 6 (Axiom 2 x
-
Removing parentheses, 2x
2(x 2
-
a;
3)
+
6
9x
Uniting,
x
Ex.
If
2.
x
= =
89),
72
-3
72
(a; 4-
Bx
4)
-
12
x.
Qx.
2z-2a;-f3# + C:E=-6-f72-12.
Transposing,
Check.
3,
= 6,
each
member
-I
reduced to
1.
14
5
Solve
is
= 64. = 6.
x
+1
x
+3
-!)(&+ 1) (x + 3), + 1) (a + 3) - 14 (a; - l)(z + 3) = - 9(se + !)( 14 x 2 - 28 x + 42 = - 9 x2 + 9. 5 x2 + 20 x + 15 15 - 42 + 9. 14 z 2 + z 2 + 20 x - 28 a = 5 x2 - 8 x = - 48, = 6.
Multiplying by (x
5(3
Simplifying,
Transposing, Uniting,
85
Check.
If
x
6,
each
member
is
reduced to
108
1.
I).
FRACTIONAL AND LITERAL EQUATIONS EXERCISE Solve the following equations
^
3
109
58
:
_ +7 a?
4
-
32
'2
--. 3
a?
4
"T"" 4
"
2
4- 1
_7-7
o
""~TiT"
a?
a;
'
3
3 10.
^-1 = 9. a/
-
-
12.
4
5
= 12.
+4 14.
1+5
= 19
-^^0
= 5.
&
a/
'
1
11.
- = 2.
a:
16.
= xx a?
a?
1
*>
7
a;
13.
= 2.
a?
18.
hi-
15.
x
+^ + 3 = 11. ^'
2,
4
a;
20
on
y
334
+2
y-2
+1
y-3 ==
2 ^
16
ELEMENTS Of ALGEBRA
110
24.
?_=_.
29.
y+3~2
25.
l-~.
26 26.
4a4-l4* +
27
.
2^12 = 2
28
.
= 34.
35.
36.
31 31.
.
32
.
3
_J_ = _J3 ._ _
-
20
x+3 x-3
33.
.
2
6 .
-
2
o^-
-
13
J_.
.
3x
3
3x-2
3x-2
3x*-2x 23 x
51
+
2a?-3 A*
1
1
22
26
4^-9
2^4-3 37.
-
38
=
7
^^
'
39
'
x- 11_4 x-
40.
149. If two or more denominators are monomials, and" the remaining one a polynomial, it is advisable first to remove the monomial denominators only, and after simplifying the resulting equation to clear of all denominators.
FRACTIONAL AND LITERAL EQUATIONS Ex.1.
Solve
5#
10 Multiplying each term by tors,
16 x
Transposing and
5
1
L. C. M. of the monomial denomina-
10, the
2( +3-~
~ &Q n =:
x
a;
26
1,
a;
-
5
x
:
a;
=
-
20 g
~ Jff
each
9,
member
Solve the following equations 41
is
2.
.
1
a:
= 20 x -
5x
Dividing, If
16 x
5
Transposing and uniting,
Check.
=
5
uniting.
Multiply ing by 6
60.
45.
=
9.
reduced to
^.
:
5a;-2
42 9
,,
43.
24
a;
-f
8#-f 2__ 2x
13
15
44.
5
7
~~7-16*
10 x -f 6 __ 4a;-r-7 5
6a?
6a-fll~~
+l
3
6x-flO '
5
2a?~25
15
28
64-14
17a?~9 14
~
-
.2
18
3
==
9
111
7a;-29 507-12'
ELEMENTS OF ALGEBRA
112
LITERAL EQUATIONS 150. Literal equations ( 88) are solved by the same method as numerical equations. When the terms containing the unknown quantity cannot be
actually added, they are united
factoring.
= (a -f 6) mnx = (1 4- m
ax
Thus,
by
-f-
x -f- m 2*
bx
jr.
2
mn) x.
Ex.1.
a
a
b
ax-
Clearing of fractions,
a
z
+ bx ax
Transposing, Uniting,
6)z
l
= !=?_=^6?
a;
=a
-f
-f
(a
= 3 & 2 ab. = 2 -f b 2 - 3 6 2 = a' - & - 2 62
IP
bx
Dividing,
Reducing
151.
It
a to
lowest terms,
frequently occurs that the
expressed by
Ex.
2.
or
x, y,
L=
If
3a-c
Multiplying by 3 (a
a
=
Transposing
all
, ?
Uniting the Dividing, 5>
a,
fr
unknown
not
letter is
find
a
in
terms of b and
c)
ac
+6
= (2a + &)(3a-c). = 6 a2 - 2 ac + 3 aft - be.
6c
terms containing a 6 ab
Simplifying,
-
-f 6.
3(a-c)
- c) (3 a -
6 a&
2
.
z.
6(rt-fc)(a-c) 6 a2
ab.
2
6 ac
to
one member,
+ 2 ac
3 ab
9 a&
4 ac
and multiplying by
1,
= =
a(9 b
= -l^ 9 b
4-
4 c
6 6c
~
5c.
7 6c. -f
4 c)
= 7 &c.
c.
FRACTIONAL AND LITERAL EQUATIONS EXERCISE Solve the following equations
59
:
*,
2. 3.
4.
6.
+ 3a; = 8 4 #. a + 26+3aj=2o + 6 + 2a?. mx = n.
11.
ax
13. 14.
-4-- = c
21.
a
.
Z>
3(*-
12.
10.
_
iw
= 3 (6 a). = 2(3a = aaj-ffta? + 7^ = 0*+^
8.
9.
113
4 (a
1
x)
3(2a +
aj)
a).
25
1
a;
?+l a/'~~
= 2L
-f- a;
l
1
.
;i
c.
^
^o;
-f-
+ &o; = 6 (m -f n) = 2 a + (m-?i)a?.
a^
co?.
a?
n) x
15.
(wi
16.
- = H.
18.
-
=px + q.
- = n.
17.
+ = xx
1.
*
*
-
26.
m
= 5.
a?
x!7
-
x
a ITo
n
IIL
x
b
,
T
= vt, = rt, s =
_ ~
2 8.
If s
solve for v.
29.
If s
solve for
30.
If
31.
If
V-t
2
If
Solve the same equation for^).
34.
The formula
for simple interest
-, solve for a. c
-=-+!,
f P
33.
solve for y a.
^^a = 1
32.
,
(
t.
solve for/.
q
30, Ex. 5) is
t
=^,
denoting the interest, p the principal, r the number of $>, and n the number of years. Find the formula for: i
() The (6) (c)
The The i
principal. rate.
time, in terms of other quantities.
ELEMENTS OF ALGEBRA
114
(a) Find a formula expressing degrees of Fahrenheit terms of degrees of centigrade ( has invested $ 5000 They both derive the same income from their How much money has each invested ?
20. An ounce of gold when weighed in water loses -fa of an How many ounce, and an ounce of silver -fa of an ounce. ounces of gold and silver are there in a mixed mass weighing
20 ounces in 21.
A can
22.
A
air,
and losing
1-*-
ounces when weighed in water?
do a piece of work in 3 days, and B in 4 days. In how many days can both do it working together ? ( 152, Ex. 2.) can do a piece of work in 2 days, and
how many days can both do
it
B in 6 days.
working together
In
?
23. A can do a piece of work in 4 clays, and B In how many days can both do it working together
in ?
12 days.
ELEMENTS OF ALGEBRA
118
The
and their solutions differ only two given numbers. Hence, by taking for these numerical values two general algebraic numbers, e.g. m and n, it is possible to solve all examples of this type by one example. Answers to numerical questions of this kind may then be found by numerical substitution. The problem to be solved, therefore, is A can do a piece of work in m days and B in n days. In how 153.
last three questions
in the numerical values of the
:
many days we
If
let
method of
can both do
x
= the
it working together ? required number of days, and apply the
we
170, Ex. 2,
Solving, 3;=
m
-f-
obtain the equation
n
Therefore both working together can do
To
-- = -. n x
m
it
in
mn
m
-f-
n
days.
A can do this work in 6 days Q = 2. and n = 3. Then
find the numerical answer, if
ft
and
B
in 3 days,
they can both do
make it
m
6
i.e.
6
in 2 days.
Solve the following problems 24.
In
piece of
how many days
work
if
can
(a) (6) (c)
(d)
3
:
A
and
each alone can do
ofdavs:
I
A in 5, A in 6, A in 4, A in 6,
it
B
working together do a
in the following
number
B in 5. B in 30. B in 16. B in 12.
25.
Find three consecutive numbers whose sum
is 42.
26.
Find three consecutive numbers whose sum
is 57.
The
last
two examples are
special cases of the following
problem 27. Find three consecutive numbers whose sum equals m. Find the numbers if m = 24 30,009 918,414. :
;
;
FRACTIONAL AND LITERAL EQUATIONS
119
Find two consecutive numbers the difference of whose
28.
squares
is 11.
Find two consecutive numbers -the difference of whose
29.
squares
is 21.
30. If each side of a square were increased by 1 foot, the area would be increased by 19 square feet. Find the side of the square.
The
last three
examples are special cases of the following
one:
The
31.
difference of the squares of
two consecutive numbers
find the smaller number.
By using the result of this problem, solve the following ones Find two consecutive numbers the difference of whose squares is ?n
;
:
is (a)
51, (b) 149, (c) 16,721, (d) 1,000,001.
Two men
same hour from two towns, 88 one traveling 3 miles per hour, and the second 5 miles per hour. After how many hours do they meet, and how many miles does each travel ? 32.
miles
start at the
apart, the
first
Two men start at the same time from two towns, d miles the first traveling at the rate of m, the second at the apart, After how many hours do they rate of n miles per hour. 33.
meet, and how many miles does each travel ? Solve the problem if the distance, the rate of the
first,
and
the rate of the second are, respectively (a) 60 miles, 3 miles per hour, 2 miles per hour. 2 miles per hour, 5 miles per hour. (b) 35 miles, :
(c)
64 miles, 3J miles per hour,
4J-
miles per hour.
by two pipes in m and n minutes how In many minutes can it be filled by the respectively. two pipes together ? Find the numerical answer, if m and n are, respectively, (a) 20 and 5 minutes, (b) 8 and 56 minutes, 34.
(c)
A cistern can
6 and 3 hours.
be
filled
CHAPTER X RATIO AND PROPORTION 11ATTO
The
154.
Thus the
two numbers number by the
ratio of
dividing the
first
ratio of a
and
b
is
is
the quotient obtained by
second.
- or a *
b.
The
ratio is also frequently
b
(In most European countries this symbol is employed as the usual sign of division.) The ratio of 12 3 equals 4, 6 12 = .5, etc.
written a
:
the symbol
b,
:
A
155.
:
being a sign of division.
:
ratio
is
used to compare the magnitude of two
numbers. " a Thus, instead of writing
a
:
b
is
6 times as large as
?>,"
we may
write
= 6.
156.
The
first
term of a ratio
is
the antecedent, the second
term the consequent. In the ratio a
:
ft,
a
the
is
numerator of any fraction
is
The
antecedent, b is the consequent. the antecedent, the denominator
the
consequent.
157.
The
ratio -
is
the inverse of the ratio -.
a 158.
Since a ratio
fractions if its
may
b is
a
fraction, all principles
be af)plied to ratios.
E.g. a ratio
is
relating
terms are multiplied or divided by the same number,
Ex.
1.
Simplify the ratio 21 3|.
A somewhat shorter way
:
would be to multiply each term by 120
to
not changed
6.
etc.
AND PROPORTION
RATIO Ex.
2.
Transform the
ratio 5
1.
equal
5
*~5
3J so that the
:
33 :
~
72:18.
3.
J:l.
4.
$24: $8.
4|-:5f
6.
5 f hours
3:4.
9.
8.
3:1}.
10.
8^-
hours.
4
.
27 06: 18 a6.
11.
16 x*y
12.
64 x*y
:
:
24
xif.
48
a-y
3 .
ratios so that the antecedents equal
:
16:64.
15.
159.
|
:
:
7|:4 T T
Transform the following
two
:
5.
7.
unity
61
ratios
62:16.
Simplify the following ratios
term will
'4*
EXERCISE
1.
first
3
5
Find the value of the following
2.
121
A
16.
17.
7f:6J,
is
proportion
:
1.
18.
16a2 :24a&.
a statement expressing the equality of
ratios.
= |or:6=c:(Z are
160.
The
first
proportions.
and fourth terms of a proportion are the and third terms are the means. The last
extremes, the second
term
is
the fourth proportional to the
first three.
In the proportion a b = c c?, a and d are the extremes, b and c the means. The last term d is the fourth proportional to a, b, and c. :
:
If the means of a proportion are equal, either mean the mean proportional between the first and the last terms, and the last term the third proportional to the first and second
161.
is
terms. In the proportion a b :
and
c,
and
c
is
=
b
:
c,
b is the
mean
the third proportional to a and
b.
proportional between a
ELEMENTS OF ALGEBRA
122 162.
Quantities of one kind are said to be directly proper
tional to quantities of another kind, if the ratio of any two of the first kind, is equal to the ratio of the corresponding two
of the other kind. ccm. of iron weigh ,30 grams, then G ccm. of iron weigh 45 grams, 6 ccm. = 30 grams 45 grams. Hence the weight of a mass of iron is proportional to its volume. " we " NOTE. Instead of u If 4
or 4 ccm.
:
:
may say,
directly proportional
pro-
briefly,
portional.'*
Quantities of one kind are said to be inversely proportional to quantities of another kind, if the ratio of any two of the first kind is equal \o the inverse ratio of the corresponding two of
the other kind. If 6 men can do a piece of work in 4 days, then 8 men can do it in 3 days, or 8 equals the inverse ratio of 4 3, i.e. 3 4. Hence the number of men required to do some work, and the time necessary to do it, are :
:
:
inversely proportional.
163. In any proportion product of the extremes.
t/ie
a
Let
:
product of the means b
=c
:
is
equal
to the
d,
!-; Clearing of fractions, 164.
ad =
The mean proportional
the square root
be.
bettveen two
numbers
is
equal to
of their product.
Let the proportion be
Then Hence
=b = ac.__(163.) b = Vac.
a b :
6
:
c.
2
165. If the product of two numbers is equal to the product of two other numbers^ either pair may be made the means, and the
other pair the extremes, of a proportion. If
mn = pq, and we
divide both
?^~ E. q~~ n
(Converse of
members by
nq,
163.)
we have
AND PROPORTION
PATIO Ex.
Find
1.
x, if
6
:
x = 12
Ex.
rn
a?
8:6 =
4$ = 35,
If
a
6
a
166. I.
III.
7
=c
d,
6
:
=d
:
:
c.
= 35
;
7
(Called Alternation.)
:
:
of these propositions
To prove
may be proved by
is
true
example
ad
if
-
Division.)
a method which
:
d
b
= be = be. ad = be. bd
bd.
ad
if
But
163.)
(
^ =^'
Hence
d
o
These transformations are used to simplify proportions. Change the proportion 4 5 = x 6 so that x becomes the
167.
:
last term.
By
true.
(Frequently called Inversion.)
illustrated by the following
I.
is
then
:
V.
Or
true
4|.
hence the proportion
+ b:b = c + d:d. (Composition.) a + b:a = c + d:c. d d. (Division.) a b b=c = b d. a+b a c-)-d:c (Composition and
IV.
This
:
is
a
Or
Any
:
and 5 x
a:c=b:d.
II.
is
Determine whether the following proportion
2.
t:
8 x
7.
= 42. (163.) = f f = 3 J.
12x
Hence
:
123
inversion 5
:
4
=6
:
x.
:
ELEMENTS OF ALGEBRA
124 IT. its
Alternation shows that a proportion is not altered when its consequents are multiplied or divided by
antecedents or
the same number. E.g. to simplify
consequents by
3:3
Or
1:1
To
III.
divide the antecedents by 16, the
48:21=32:7x,
7,
= 2:3. = 2:x,
i.e.
5:6
simplify the proportion
Apply composition,
11
:
6
=4
:
5
division,
Divide the antecedents by
V. To simplify
?
5, 1
m 3n
Apply composition and
=4
x
:
x.
x.
IV. To simplify the proportion 8 Apply
= 2.
x
:
:
3 3
= =
5
:
1
:
= 5 -f x
3
:
:
x.
jr.
x.
= + *.
mx
= ^-
division,
2x
tin
.!=!*.
Or
3n
=-
Dividing the antecedents by m, JJ
n
A parenthesis is understood
NOTE.
x x
about each term of a proportion.
EXERCISE
62
Determine whether the following proportions are true 1.
2. 3.
5^:8 = 2:3. = 7:2f 3J.:J
:
= 12 5ft. 8ajy:17 = i^:l-^. 11
4. 5.
:
5
:
15:22=101:15.
Simplify the following proportions, and determine whether they are true or not :
6.
7.
10.
= 20:7. 8. = 9. 72:50 180:125. m n (m n) = (m + rif m 2
2
2
:
:
= 24:25. 13 = 5f llf
18:19
120:42
6 2
:
n 2.
:
RATIO AND PROPORTION Determine the value of x 11. 12.
:
40:28 = 15:0;. 112:42 = 10:a.
13.
03:a?=135:20.
14.
a?:15
17.
1, 3, 5.
21.
3, 3t, f.
20.
2, 4, 6.
22.
ra, w,j>.
25.
16 and
31. 35.
to
37. 38. 39.
40.
,
rap, rag.
1
and
a.
27.
29.
a and
1.
34.
ra
to
:
8 a 2 and 2 b 2
Find the
2
28.
2 a and 18
If ab
ra
a 2 and ab.
33.
x 10
23.
:
32.
Form two
:
14 and 21.
and 2/.
equation 6 36.
:
4 and 16. |-
96.
26.
Find the mean proportional 30.
:
to:
19.
28.
4z = 72
= 35:*. 4 a*:15ab = 2a:x. 16 n* x = 28 w 70 ra.
18.
Find the third proportional
:
2.8:1.6
16.
Find the fourth proportional
9 and 12.
21
15.
= l^:18.
24.
125
a.
+ landra
proportions commencing with 5 from the
= 5 x 12.
= xy,
form two proportions commencing with
ratio of
x
:
y, if
6x = 7y. 9 x = 2 y. 6 x = y. mx = ny.
41. 42. 43. 44.
+ fyx = cy. x:5 = y:2. x m = y n. 2 3 = y #. (a
:
:
:
45.
7iy = 2:x.
46.
y
:
47.
y
:
=x 1 =x
b
2:3 = 4- x:
49.
6:5
50.
a
x.
= 15-o;:ff. 2= 5 x x. :
:
a.
:
a2
.
:
Transform the following proportions so that only one 48.
b.
:
contains x:
:
1.
.
= 2 + x: x. = 3 43 + x. 5=
51.
22: 3
52.
19
53.
2
:
:
18
a?
a;
:
:
a?.
terra
ELEMENTS OF ALGEBEA
126
State the following propositions as proportions : T (7 and T) of equal altitudes are to each, othei
54.
(a) Triangles
as their basis (b
and
b').
(6) The circumferences (C and C ) of two other as their radii (R and A"). (c) The volume of a body of gas (V) is 1
circles are to
each
inversely propor-
tional to the pressure (P).
The
(d)
(A and
areas
A') of two circles are to each other as
(R and R'). The number of men (m) is inversely proportional to the number of days (d) required to do a certain piece of work. the squares of their radii (e)
55. State whether the quantities mentioned below are directly or inversely proportional (a) The number of yards of a certain kind of silk, and the :
total cost.
The time a
(b)
train needs to travel 10 miles,
and the speed
of the train.
The length
(c)
of a rectangle of constant width, and the area
of the rectangle.
The sum
(d)
of
money producing $60
interest at
5%, and
the time necessary for it. (e) The distance traveled by a train moving at a uniform rate, and the time.
A
56.
line 11 inches long
A
22 miles.
The
57.
their radii.
4
:
on a certain
areas of circles are proportional to the squares of If the radii of two circles are to each other as
and the area of the smaller
7,
what
is
58.
map corresponds to how many miles ?
line 7^- inches long represents
circle is
8 square inches,
the area of the larger?
The temperature remaining
the same, the volume of a
A
body of gas inversely proportional to the pressure. under a pressure of 15 pounds per square inch has a volume of gas
is
16 cubic
feet.
What
will be the
12 pounds per square inch ?
volume
if
the pressure
is
RATIO AND PROPORTION The number
69.
127
of miles one can see from an elevation of
very nearly the mean proportional between h and the diameter of the earth (8000 miles). What is the greatest distance a person can see from an elevation of 5 miles ? From h miles
the
is
Metropolitan
McKinley (20,000
Tower (700
feet
high) ?
From Mount
feet high) ?
168. When a problem requires the finding of two numbers which are to each other as m n, it is advisable to represent these unknown numbers by mx and nx. :
Ex. as 11
1. :
Divide 108 into two parts which are to each other
7.
Let then
Hence or
Therefore
Hence and
= the first number, = the second number. 11 x -f 7 x = 108, 18 x = 108. x = 6. 11 x = 66 is the first number, 7 x = 42 is the second number. 11
x
7
x
A line AB, 4 inches long, 2. produced to a point C, so that
Ex. is
(AC): (BO) =7: 5.
Find^K7and BO.
Let
AC=1x.
Then
BG = 5 x. AB = 2 x.
Hence
Or
2 x
=
4 '
r
i
A
B
4.
x=2. Therefore
7
=
14
= AC.
1
ELEMENTS OF ALGEBRA
128
EXERCISE 1.
Divide 44 in the ratio 2
:
2.
Divide 45 in the ratio 3
:
7.
3.
Divide 39 in the ratio 1
:
5.
4.
A line 24 inches
long
63
9.
divided in the ratio 3
is
:
5.
What
are the parts ? 5. Brass is an alloy consisting of two parts of copper and one part of zinc. How many ounces of copper and zinc are in 10 ounces of brass ?
consists of 9 parts of copper and one part of ounces of each are there in 22 ounces of gun-
Gunmetal
6.
How many
tin.
metal ?
Air is a mixture composed mainly of oxygen and nitrowhose volumes are to each other as 21 79. How many gen, cubic feet of oxygen are there in a room whose volume is 4500 7.
:
cubic feet? 8.
The
7 18.
total area of land is to the total area of
If the total surface of the earth
water as
is
197,000,000 square miles, find the number of square miles of land and of water. 9. Water consists of one part of hydrogen and 8 parts of :
oxygen.
grams
How many
grams of hydrogen are contained in 100
of water?
10.
Divide 10 in the ratio a
11.
Divide 20 in the ratio 1 m.
12.
Divide a in the ratio 3
13.
Divide
:
b.
:
m
:
7.
in the ratio x:
y
%
The
three sides of a triangle are 11, 12, and 15 inches, and the longest is divided in the ratio of the other two. How 14.
long are the parts ? 15. The three sides of a triangle are respectively a, 6, and c inches. If c is divided in the ratio of the other two, what are its
parts ? (For additional examples see page 279.)
CHAPTER XI SIMULTANEOUS LINEAR EQUATIONS 169.
An
equation of the
unknown numbers can be the unknown quantities.
first
degree containing two or more by any number of values of
satisfied
2oj-3y =
If
6,
2 y = - -. y = a?
,-L
then
(1)
5
/0 \ (2)
= 0, =,y=--|. x = 1, y = 1, etc.
x
I.e. if
If If
Hence, the equation is satisfied by an infinite number of sets Such an equation is called indeterminate.
of values.
However,
there
if
different relation
is
given another equation, expressing a y, such as
between x and *
+ = 10,
(3)
unknown numbers can be found. From (3) it follows y 10 x and since
these
y
to be satisfied
by the same values of x and
y must be equal.
Hence
2s -5
= 10 _ ^
o
the equations have the two values of
y,
(4)
= 3. is x = 7, which substituted in (2) gives y both are to be the same satisfied Therefore, equations by values of x and y, there is only one solution. The
root of (4) if
K
129
ELEMENTS OF ALGEBRA
130 170.
A
system
a group of equa by the same values of the unknown
of simultaneous equations is
tions that can be satisfied
numbers. 6 and 7 x 3y = by the values x = I, y
are simultaneous equations, for they are 2 y = 6 are But 2 x 2. 6 and 4 x y not simultaneous, for they cannot be satisfied by any value of x and y. The first set of equations is also called consistent, the last set inconsistent.
x
-H
2y
satisfied
I
171. Independent equations are equations representing different relations between the unknown quantities such equations ;
cannot be reduced to the same form.
~ 50, and 3 x + 3 y =. 30 can be reduced to the same form -f 5 y Hence they are not independent, for they express the x -f y 10. same relation. Any set of values satisfying 5 x + 6 y = 60 will also satisfy the equation 3 x -f- 3 y = 80. 6x
;
viz.
172.
A
unknown
system of two simultaneous equations containing two quantities is solved by combining them so as to obtain
one equation containing only one
unknown
quantity.
The process of combining several equations so as make one unknown quantity disappear is called elimination. 173.
174.
The two methods
By By
I.
II.
of elimination
to
most frequently used
Addition or Subtraction. Substitution.
ELIMINATION BY ADDITION OR SUBTRACTION 175.
E,X.
Multiply (1) by
Multiply (2) by
Solve
2, 3,
Subtract (4) from (3), Therefore,
-y=6x 6x
-f
-
4y
- 26.
= - 24, 26 y = 60. y = 2. 21 y
(3) (4)
SIMULTANEOUS LINEAR EQUATIONS
131
Substitute this value of y in either of the given equations, preferably
the simpler one (1),
3x
Therefore
In general, eliminate the
common
letter
coefficients
Multiply (1) by Multiply (2) by
+ 2.2 = 9 + 4 = 13, 3-7- 2 = 6- 14 =-8.
(3)
and
25 x - 15 y 39 x + 15 y
5,
3,
(4),
Therefore Substitute (6) in (1),
Transposing, Therefore
5
Check.
.
13
Hence
to eliminate
= 235. = 406. 64 x = 040. x = 10. 60 - 3 y = 47. 3y = 3. y = 1. x = 10. 10 - 3 1 = 47, 10 + 5 1 = 135.
(3) (4)
(6)
by addition or subtraction :
Multiplyy if necessaryy the equations by such
make
have the lowest
3. 8
2.
176.
whose
multiple.
Check.
Add
+ 4 = 13 x = 3. y = 2.
the coefficients
If the signs of these if unlike,
add
numbers as
will
of one unknown quantity equal. coefficients
are
like,
subtract the equations;
the equations.
EXERCISE
64
Solve the following systems of equations and check the
answers:
'
ELEMENTS OF ALGEBRA 5.
^ = ll.
13-
v
2/
= 24.
17.
-I
i
3
a;
6-1
l7a; +
7
'
1fi fl ,4.1ft
= 6.
is
+ 22/ = 40,
fl* r A
O
1
8. I
5y
oj
t
K
= 17. 19
if
/*
1i>
>
function of the
degree is an integral Y'
rational function
71
4J, etc.
193. first
,,
GRAPHIC REPRESENTATION OF FUNCTIONS EXERCISE
Draw
157
73
the graphs of the following functions:
1.
a?
+ 2.
4.
2x +
l.
7.
2-3x.
10.
a?
2.
x-l.
5.
3x
2.
8.
1
11.
xz + x.
3.
2
12.
4a?
13.
I.
a? 2
a; 2
4
+ 4.
16.
a;
the graph of
the diagram find (3.5)2;
Va25;
or
(c)
a
1.
a*
3. y = 2x = -4. a;
?/
2
ar.
from
2
#=
4 to
05
= 4,
and from
:
(ft)
(/)
(_
1.5)2;
Vl2^
;
22. Draw the graph of or from the diagram determine:
(d)
20.
-fa--
Draw
(6)
19.
6 -fa- -or.
2.
.
x+1.
2
21.
(a)
-Jar
17.
a;
(e)
2
18.
a;
15.
(a)
8
9.
a?.
-3 a -8.
14.
2
a;
6.
a?.
4
(C )
(-2.8)';
(0)
V5;
a?
+2
from x
2
(d)
(-If)
(^)
VlO-'S".
1 to
a;
= 4,
The values of the function if x = \, 1J-, 2J-. The values of a?, if a;2 4 # + 2 equals 2, 1, 1-J-. The smallest value of the function. The value of x that produces the smallest value
;
and
of the
function.
The values of x that make 2 4 a? + 2 = 0. 2 4 x -f 2 = 0. (/) The roots of the equation x 2 The of a x -f 2 = the 4 roots 1. equation (
fi
to in factors
3 a268 )
a 8 = _ (2m )
(8
+ 2 = a. = 6+ 5 + +fi =
2+2 5
____ 16 *)"" 27 n 165
(-
62.
by
ELEMENTS OF ALGEBRA
166 To find
the exponent
of the power of a power, multiply tht
given exponents.
To
raise
a product
to
a given power,
raise each of its factors to
the required power.
To
raise
a fraction
to
a power, raise
terms to the required
its
power.
EXERCISE Perform the operations indicated 1.
2.
(>y.
2 4
(-a )
5.
3.
-
76 :
2 5
(-a )
/2mV. (-277171
2 11 .
(afc )
24.
\ 3 J
4
6.
4.
.
)*.
-
'
M-W 10.
(-2ar).
27
'
'
11.
-
13.
^---
/
_4_V ' _4_
V
V/
/-2?n?A 4
30.
3
15.
am-Vy)
16.
(-|^^)
. '
2
V 3xy )'
.
INVOLUTION OF BINOMIALS 209.
210.
by
and
+
The
square of a binomial
The
cube of a binomial
&-
we
was discussed
63.
obtain by multiplying (a
= a + 3a 6 + 3a6 + * 6) = a - 3 a 6 -f 3 a6 - 6 (a 6)
(a
in
3
3
2
2
8
8
3
2
2
8
_j_
,
.
+ 6)
1
INVOLUTION Ex.
Ex.
167
Find the cube of 2 x -f- 3 y.
1.
=
(2s)
s=
8 a; 3
3 + 3(2aO*(Sy) + 3(2aj)(3y)> + 36 z2y + 54 xy* + 27 y3 .
n of 3 x* - y
Find the cube
2. 2
(3 x
.
- y) = (3 y?y - 3(3 a*)a(y = 27 a - 27 ay + 9 x y2n 6
2
EXERCISE
77
Perform the operations indicated: 1.
2.
(a
+ &)8
(a?-?/) 3
(a-fl)
4.
(m-2) 8 (w+w)
6.
8.
.
3.
5.
7.
.
8
.
3
(a-j-7)
Find the cube root of
3
+ 4aj)
(7 a
-
-I) 2
.
a;)
(l
8 .
3
-I)
.
3 .
+5a)
3 8
(1
.
8
13.
(3a-f26)
14.
(6m+2w)
15.
(3
a- 6
16.
(3a
17.
(a
18.
(4
2
or*
62
.
8 .
8 ft)
-l)
.
3 .
-
:
20. 21.
a8 -3a2 + 3a-l.
a
.
10.
+ 3a 6 + 3a& -f-& ^-Sx^ + S^ -^
19.
2
3
+
3
a
12.
.
(1
2
(3
lx
-
-a)
9.
.
8
(5
2
3
2
3
86
23.
.
w + 3 w + ra8 -126 + G6-l. 2
1 -f 3
22.
.
3
.
2
211. The higher powers of binomials, frequently called ex. pansions, are obtained by multiplication, as follows :
+ 6) = o + 3 d'b + 3 a6 + = a + 4 a?b + 6 a & + 4 a6 + b + (a 6) = a + 5 a 6 + 10 a*b + 10 a 6 -f 5 aM + 6 (a + 6) 8
8
4
4
5
5
2
b*.
(a
2
2
4
3
.
4
2
2
An
examination of these results shows that
1.
The number of terms
is
s
3
,
etc.
:
1 greater than the exponent of the
binomial. TJie exponent of a in the first term is the same as the expo2. nent of the binomial, and decreases in each succeeding term by L
ELEMENTS OF ALGEBRA
168 T7ie
3.
exponent ofb
1 in the second term of the result,
is
and
increases by 1 in each succeeding term.
The The
4. 5.
of the first term is 1. of the second term equals the exponent of the
coefficient coefficient
binomial 6. TJie coefficient of any term of the power multiplied by the exponent of a, and the result divided by 1 plus the exponent of b, is the coefficient of the next term.
Ex.
Expand
1.
=
5
ic
-f
5 x*y
(x
+
10
^V +
Ex.2. Expand (a??/)
+
10 x*y*
The
212.
+ y5
.
5 .
2
x5
5 xy*
+
x'2
10
(-
+
5 x4 y
signs of the last answer arc alternately plus y are positive, and the
minus, since the even powers of
and odd
powers negative. Ex.
Expand
3.
i?i 2
4
mn
15. (l
.
8
11.
13.
6 2 ) 5.
8 .
-f c)*.
.
5
(mnp
.
I)
2
5
20.
(2w
21.
(3a -f5)
.
-f-l)
2
4
2 22. (2 a
4
.
.
5)
23.
(2a-5c)
24.
(1 -f 2
4
4 a:)
.
.
CHAPTER XIV EVOLUTION 213. tity
;
is the operation of finding a root of a quan the inverse of involution.
Evolution
it is
\/a
=
V
27
\/P 214. 1.
x means x n
=y
= x means
It follows
Any
means r'
= y
?>
a.
=
= 6-,
27, or y
or x
~
3.
&4 .
from the law of signs
in evolution that
even root of a positive, quantity
may
:
be either 2wsitive
or negative. 2.
Every odd root of a quantity has
the
same sign as
the
quantity.
V9 = +
3,
or
-3
(usually written
\/"^27=-3, (_3) = -27. and ( v/o* = a, for (+ a) = a \/32 = 2, etc.
3)
;
for (-f 3) 2
and
(
3)
2
equal
0.
for
4
4
,
a)
4
= a4
.
215. Since even powers can never be negative, it is evidently impossible to express an even root of a negative quantity by Such roots are called imaginary the usual system of numbers.
numbers, and
all
other numbers are, for distinction, called real
numbers. Thus
V^I is an imaginary number, which can be simplified no further. 109
ELEMENTS OF ALGEBRA
170
EVOLUTION OF MONOMIALS The following examples root
are solved
by the
definition of a
,
:
=
Ex.1.
v/^i2
Ex.
2.
3/0**
Ex.
3.
v^SjW 3 = 2 a
= am
= ^/gL^g * c*
Ex
A
82
for (a")"
,
Ex.4.
5
(a
a*, for
a
3
)*
= a 12
= a mn
&c*, for (2
a"
.
.
a 2 6c4 ) 8
=
2
To
216.
?*-
= .lL,for(*Siy 3 3 6 c* \ c*J
extract the root
2 b'
of a power, divide
ft^c20
243
the exponent
by the
index.
A root of a product equals the product of the roots of the factors. To extract a root of a fraction, extract the and denominator. Ex.
6.
\/18
.
14
63
= V2 3* = 2 32 6
25
.
.
Ex.
Ex.
7.
8.
VT8226
= V25
Find (x/19472)
Since by definition
Ex.
729
2 .
7
.
7
.
.
roots of the
82
.
62
= V2*
.
numerator
3i
.
6-
= 030.
7
= V26TIT81 = 5-3.9 = 136.
2 .
= a, we
( v^)"
have (Vl472) 2
= 19472.
9.
= 199 + (_ 198) - 200 - (- 201) = 2. EXERCISE 5
5
1.
-v/2
.
3.
-fy
2.
V?.
4.
-v/2^.
9.
V36
9
-
100
a
2 .
3 .
5.
V5
6.
-v/2
79 2
7 2.
3
33
10.
V25
7.
53
\/2
9
4
16.
v- 125- 64
8.
.
4
9
5
4 .
EVOLUTION
33.
34.
35.
36.
VH) + (Vl9) 2
(
2
(
VI5) x ( VT7)
2
2
171
- (V200) -f ( VI5)
2
(V2441) ~(V2401)
2
r
+ b\
28.
-\/d -\-Vab
29.
V8- 75- 98- 3.
30.
V20
31.
V5184.
32.
V9216.
2
-f
(
V240)
x ( V3)
.
45
9.
2 .
3 .
2 .
2
(Vl24) -{
EVOLUTION OF POLYNOMIALS AND ARITHMETICAL
NUMBERS
A
217.
trinomial is a perfect square if one of its terms is
equal to twice the product of the square roots of the other terms. In such a case the square root can be found ( 116.)
by inspection. Ex.
1.
Find the square root of a2 - 6 ofy 2 -f 9 y4
_ 6 ary -f 9 y = (s - 3 y2) ( vV - 6 tfif + 9 y = O - 3 ;/). 4
a*
Hence
8
.
4
EXERCISE
.
2
116.)
3
80
Extract the square roots of the following expressions 2
1.
a -f2
2.
l
+ l.
2y-h2/
2 .
3.
^-40^4- 4/.
5.
4
9^ + 60^ +
6.
-
2 2/ .
:
ELEMENTS OF ALGEBEA
172 7.
4a2 -44a?> + 121V2
10.
8
4a
+ 6 + 4a&.
11.
49a 8 -
12.
16 a 4
.
.
s
2
mV-14m??2)-f 49;>
9.
13.
2 .
- 72 aW + 81 &
4 .
#2
14. 15.
a2
-
16.
a2
+ & + c + 2 a& - 2 ac - 2 &c. 2
2
218. In order to find a general method for extracting the square root of a polynomial, let us consider the relation of a -f- b 2 2 to its square, a -f- 2 ab + b .
The
first
term a of the root
the square root of the
is
first
2
term
a'
.
The second term
of the root can be obtained
second term 2ab by the double of
2ab
a-\-b
is
the root
if
a,
by dividing the the so-called trial divisor;
,
the given expression is a perfect square. it is not known whether the given
In most cases, however,
expression is a perfect square, and b (2 a -f b), i.e. the that 2 ab -f b 2
=
and
b,
we have then to consider sum of trial divisor 2 a,
multiplied by b must give the last two terms of the
square.
The work may be arranged
as follows
a 2 + 2 ab 2
:
+ W \a + b
EVOLUTION Ex.
1.
173
Extract the square root of 1G 16x4
x*
- 24 afy* -f 9 tf.
__
10 x*
Arrange the expression according to descending powers root of 10 x 4 is 4 # 2 the lirst term of the root. 2 Subtracting the square of 4x' from the trinomial gives the remainder '24 x'2 + y. By doubling 4x'2 we obtain 8x2 the trial divisor. 24# 2 y 3 by the trial divisor Dividing the first term of the remainder, 8 /-, we obtain the next term of the root 3 y 3 which has to be added to 2 the trial divisor. Multiply the complete divisor Sx' 3y 3 by Sy 8 and subtract the product from the remainder. As there is no remainder, Explanation.
The square
of x.
,
*/''
,
,
,
,
,
4 x2
3
?/
8 is
the required square foot.
219. The process of the preceding article can be extended to polynomials of more than three terms. We find the first two terms of the root by the method used in Ex. 1, and consider Hence the their sum one term, the first term of the answer.
double of this term find the next
Ex.
2.
is
new
the
term of the
root,
by division we
trial divisor;
and so
forth.
Extract the square root of
16 a 4
- 24 a + 4 -12 a + 25 a8 s
.
Arranging according to descending powers of 10 a 2.
4
-
a.
3
+
24 a 3
4-
a2
-f
10 a 2
24 a
25 a 2
-
12 a
+4
-
12 a
+4
10 a 4
Square of 4 a First remainder. First trial divisor, 8 a 2 . First complete divisor, 8 a 2
8
Second remainder. 6 a. Second trial divisor, 8 a 2 Second complete divisor, 8 a 2
As
there
is
a.
\
a
-f 2.
no remainder, the required root
is
(4
a'2
8a
+
2}.
ELEMENTS OF ALGEBRA
174
EXERCISE
81
Extract the square roots of the following expressions
2a + a4
+ 1.
2.
3 a2
3.
a4
4.
+ 81 a 4-54 a + 81. 25 m 20 w + 34 m - 12 m 4- 9. 4-12 a& -f 37 a' 6 - 42 a -f 49 a 6
5. 6.
2 a3
x2
2 or 4-1 3
4-
16 a4
-|-
2x.
24 a3
2
3
4
2
J
2
3
3
4
4
>
4
40 afy 4-46 x
2
24 a^
8
25 x
8.
16x6 4- 73a4 4-40^4-36^4-60^.
9.
l
-f-
if 4-
4.2^4-3^4-2^ 4-
a;
4-
.
4
7.
9
.
i/
4 .
10.
1 4- 4 x 4- 10 x2 4- 20 o 4- 25 x 4 4- 24
11.
36a 4-60a 4-73a 4-40a 4-16a
12.
36it-
13.
6
4-36^?/4-69a;V4-30^4-25^ 4m 4- 12m 5 4- 9m 4 20m3 30m 4- 25.
14.
49 a 4
6
5
3
4
6
.
- 42 a*& 4- 37 a ^ - 12 a6 2
2
2
13#4 4-13ar 4-4a;6 - 14^4-4
4 0^4- 20
or
16 x
3
3
4
17.
ic
18.
729 4- 162 a2 60 a10 4- 73 a8
4-?/ 4-2x-
20.
46 a
22
16
4
4-
a?
2
4a;
XT
x*y
6 a5 4- a 6 4-
-f
_^ + 2JX
24.
44 a
8
2xif
j/
36 a
25 a
2
12
4-
-h
4-
4 64
.
4- 16.
4^
J
4
6
iK .
4
16.
4-
16
2
x
.
4-
.
6
15.
19.
5
or
2
.
- 54 a 40 a
12 a
12^.
2
6
4-
3
4-
9 a4 .
4-
16 a4
4
4-
.
25 a6 4- 40 a
:
EVOLUTION The
220.
175
square root of arithmetical numbers can be found to the one used for algebraic
by a method very similar expressions.
Since the square root of 100 is 10; of 10,000 is 100; of 1,000,000 is 1000, etc., the integral part of the square root of a number less than 100 has one figure, of a number between 100 and 10,000, two figures, etc. Hence if we divide the digits of the number into groups, beginning at the
and each group contains two digits (except the last, which may contain one or two), then the number of groups is equal to the number of digits in the square root, and the square root of the greatest square in units,
group is the first digit in the root. Thus the square root of 96'04' two digits, the first of which is 9 the square root of 21'06'81 has three digits, the first of which is 4.
the
first
consists of
Ex.
1.
;
Find the square root of 7744.
From
the preceding explanation it follows that the root has two digits, the first of which is 8. Hence the root is 80 plus an unknown number, and we may apply the method used in algebraic process.
A will
comparison of the algebraical and arithmetical method given below identity of the methods.
show the
7744 80 6400 1
160
+ 8 = 168
+8
1344
1344 Since a
Explanation.
The is
trial divisor
2 a
=
= 80,
160.
a 2 = 6400, and the first remainder is- 1344. Therefore 6 = 8, and the complete divisor
168.
As
8
Ex.
x 168
2.
=
1344, the square root of 7744 equals 88.
Find the square root of 524,176. a f>2'41 '70
2 a
a2 = +6=
41)
1400
+ 20 = 1420
00 00
341 76
28400 4
=
1444
57 76
6776
[700
6
c
+ 20 + 4 = 724
ELEMENTS OF ALGEKRA
1T6
off groups in a number which has decimal begin at the decimal point, and if the righthand group contains only one digit, annex a cipher.
221.
places,
In marking
we must
Thus the groups
in .0961
are
'.GO'61.
The groups
of 16724.1 are
1'67'24.10.
Ex.
3.
Find the square root of
6.7 to three decimal places.
12.688
6/.70
4
45 2 70 2 25
508
4064 6168 41)600
41344 2256
222.
Roots of common fractions are extracted either by divid-
ing the root of the numerator by the root of the denominator, or by transforming the common fraction into a decimal.
EXERCISE Extract the square roots of
:
82
EVOLUTION Find
177
to three decimal places the square roots of the follow-
ing numbers: 29.
5.
31.
.22.
33.
30.
13.
32.
1.53.
34.
37.
Find the
side of a square
1.01. J-.
35.
T\.
36.
JT
.
whose area equals 50.58 square
feet.
38.
Find the side of a square whose area equals 96 square
yards. 39. feet. TT
Find the radius of a (Area of a
circle
circle
whose area equals 48.4 square when R = radius and
1 equals irR ,
= 3.1410.) 40.
Find the mean proportional between 2 and
11.
CHAPTER XV QUADRATIC EQUATIONS INVOLVING ONE UNKNOWN QUANTITY
A
223.
quadratic equation, or equation of the second degree,
an integral rational equation that contains the square of 4x the unknown number, but no higher power e.g. x 2 7, 6 y2 = 17, ax 2 + bx + c = Q. is
;
A
224. complete, or affected, quadratic equation is one which contains both the square and the first power of the unknown
quantity.
A pure,
225.
or incomplete, quadratic equation contains only
unknown quantity. + bx -f c r= is a complete quadratic ax 2 = m is a pure quadratic equation.
the square of the axt
The
226.
absolute term of an equation
does not contain any In 4 x 2
7
equation.
x
-f
12
=
unknown
is
the terra which
quantities.
the absolute term
/
is 12.
PUKE QUADRATIC EQUATIONS
= a,
A pure quadratic is solved by reducing it to the form and extracting the square root of both members.
Ex.
1.
227. 2 ic
Solve 13 x2 -19
Transposing,
= 7^ + 5. 6#2 =
etc.,
x*
Dividing,
24.
= 4.
Extracting the square root of each member, x = + 2 or x
=2.
This answer Check.
frequently written x
is
13(
2)2
-
19
= 33
;
178
=
2.
7(
2)*
+
5
= 33.
179
QUADRATIC EQUATIONS Ex.2.
Solve
.=g x2
Clearing of fractions, ax
4 a2
Transposing and combining,
+ 4 ax = ax + 4 a 2 + x2 -f 2 x2 = 8 a 2 4 a2 x2 = x = V 4 a2 x= x = .
2, Dividing by Extracting the square root,
.
,
or
Therefore,
EXERCISE Solve the following equations 1.
2.
3.
-7 = 162. 0^ + 1 = 1.25. 19 + 9 = 5500. o;
2
2
a;
7. 8.
9.
10.
(a?-
6(--2)=-10(aj-l).
-?
x
+
s-3
oj
+3
= 4.
2 4fc -5'
=:
18. '
y?
b*
b
83
:
4.
16^-393 = 7.
5.
15^-5 =
6.
4 ax,
ELEMENTS OF ALGEBRA
180 on
__!_:L
a;
&
-{-
23.
If a 2 4- b 2
24.
If s
=
If
= Trr
25.
a
27.
If 2
28.
If 22
2
22
.
'
c#
=c
2
('
,
2 ,
2 ,
= 4w
2
If s
26.
r.
-f c
2
(
2a
and
-f-
1
c.
,
= 4 Trr
2 ,
solve for
r.
m.
sol ve for
G=m m '
solve for v.
If
29.
,
g
EXERCISE 1.
+a
.
solve for
= ~^-,
x
find a in terms of 6
solve for
-f 2 b*
9
4,
a
a;
solve for d.
84
Find a positive number which
is
equal to
its
reciprocal
144). 2.
A
number multiplied by
its fifth
part equals 45.
Find
the number. 3.
150. 4.
The
ratio of
two numbers
Find the numbers.
(See
2
is
:
3,
and their product
is
108.)
Three numbers are to each other as 1 Find the numbers. is 5(5.
:
2
:
3,
and the sum
of their squares 5.
The
sides of
two square
fields are as
3
Find the side
tain together 30G square feet.
:
5,
and they con-
of each field.
6. The sides of two square fields are as 7 2, and the first exceeds the second by 405 square yards. Find the side of each :
field.
228.
A
right triangle is a triangle,
_____ b
contains
c
one of
The side right angle. opposite the right angle is called the hypotenuse (c in the diagram). If the hypotenuse whose angles
is
a
units of length, and the two other sides respectively
a and b units, then Since such a triangle tangle, its area contains
c
2
=a
may
2
-f-
b2
.
be considered one half of a
square units.
rec-
181
QUADRATIC EQUATIONS The hypotenuse
7.
of a right triangle
other two sides are as 3
4.
:
Find the
is
35 inches, and the
sides.
8. The hypotenuse of a right triangle is to one side as 13:12, and the third side is 15 inches. Find the unknown sides and the area.
The hypotenuse
9.
two
The area
10.
sides are as 3
:
4.
of a right triangle is 2,
Find these
sides are equal.
and the other
sides.
of a right triangle Find these sides.
is
24,
and the two smaller
11. A body falling from a state of rest, passes in t seconds 2 over a space s yt Assuming g 32 feet, in how many seconds will a body fall (a) G4 feet, (b) 100 feet?
=
The area $
12.
the formula
whose radius equals r is found by Find the radius of circle whose area S
of a circle
= Trr
/S
=
.
-J-
2 .
equals (a) 154 square inches, (b) 44 square feet. 7r
=
-2
(Assume
2
7
13.
.)
Two
circles together contain
radii are as 3
:
Find the
4.
3850 square
feet,
and their
radii.
8 = 4 wr2 Find 440 the radius of a sphere whose surface equals square yards. 14.
If the radius of a sphere is r, its surface
(Assume
=
ir
.
-2 2
7
.)
COMPLETE QUADRATIC EQUATIONS 229.
ample
Method
of completing the
illustrates the
method
The following
square.
ex-
of solving a complete quadratic
equation by completing the square. Solve
- 7 x -f 10 = 0. x* 7 x=
or
Transposing,
10.
member can be made a complete square by adding 7 x with another term. To find this term, let us compare x 2 The
left
the perfect square x2 of
or
2m, we have
m = |.
to
add
2
(|)
mx -f m
2
Hence ,
to
2 .
Evidently 7 takes the place 7x a complete square
make x2
which corresponds
to
m
2 .
ELEMENTS OF ALGEBRA
182
Adding
2
( J)
to each
member,
Or
= f. = \ # = ff. or x = 2.
(*-i) x
Extracting square roots,
Hence x
Therefore 62
Check.
Ex.1.
-7
5
5
-|
+ 10 = 0,
22
-7
.
2
+ 10 =0.
80^69^-2 =
Dividing by
= 6. = | x |.
9 x2
Transposing,
sc
9,
Completing the square
(i.e.
15 x 2
Extracting square roots,
at
Transposing, Therefore,
Hence
Q) 2
adding
to each
(*~8) a =
Simplifying,
230.
2
member),
.
= x-\ = 2, |
\.
a;
or
J. J.
to solve a complete quadratic
:
Reduce the equation to the form x*-\-px==q. Complete the square by adding the square of one half the coefficient of x.. Extract the square root and solve the equation of the first degree thus formed.
Ex.2.
x
a Clearing of fractions,
x2
x x2
Transposing, Uniting,
s
a
+ 2 a2
-f
a
x
2 ax
- x(l
-f 2 o)
= 2 ax. 2 a*
a.
= - 2 a2 - a,
QUADRATIC EQUATIONS
183
Completing the square,
Simplifying,
Extracting square root, x
- 1+2?= "*"
-
-
Vl - 4
a2
Transposing,
x
= l+ * a
~
Therefore
*
= 1 +2
Vl
EXERCISE
85
-8o;
25.
4-2a; a=:i^-^.
26.
x(x
27
= 14.
+ 100;= 24. + 10 a = 24.
14.
7.
a?-10a=:-24.
15.
5 = 0. 3^ 25^ + 28 = 0. + 9 -f 20 x = 0. 4or + 18a -f 8a;:=0. 3# y 5 = 0. 3^ = 0(110-6). 0(0-2) = 7(0-2).
8.
aj(
16.
(5
-|-6 2. 3.
4.
0^
+
21
= 10
a?.
ar'-Sa^ -12. a* 10a=24.
+ 8=s:
7.
9.
10.
11. 12. 13.
2o3 -f9a; 2
3
or
a;
}
2
2
or
ELEMENTS OF ALGEKRA
188 f
17.
tt(3tt
18.
uz + u
+ 7tt)=6tt. 2.
21. 22. (2a?
3) (a 24. 25.
26.
ara +
(a
19.
w(w
20.
x2
2
w)=6tt. a 2 =(x
+ 2)=
(+ 3)(a?+2).
23.
3
or
-a -2
(y( j_ ?
ft
a)b.
+ 1) (a- 3) = (s + l) (3 -a).
+ c*.
27.
50.
'-3a!J -
2
a?
QUADRATIC EQUATIONS Form
the equations whose roots are
51.
3,1.
52.
3,
-4.
53.
-2, -5.
55.
54.
0,9.
56.
189
:
-2,3.
57.
1,2,3.
-2,3,0.
58.
2,0, -2.
1,
PROBLEMS INVOLVING QUADRATICS Problems involving quadratics have
in general two answers, but frequently the conditions of the problem exclude negative or fractional answers, and consequently many prob-
235.
lems of this type have only one solution.
EXERCISE
A
1.
88
number increased by three times
its reciprocal
equals
Find the number.
6J. 2.
Divide CO into two parts whose product
3.
The
difference of
of their reciprocals is
|.
two numbers is 4, and the difference Find the numbers.
Find two numbers whose product
4.
is 875.
is
288,
and whose sum
is 36.
The sum
5.
What
85. 6.
of the squares of
are the
numbers
The product
of
two consecutive numbers
is
?
two consecutive numbers
is
210.
Find
the numbers. 7.
Find a number which exceeds
8.
Find two numbers whose difference
product 9.
its
square by is
G,
-|.
and whose
is 40.
Twenty-nine times a number exceeds the square of the 190. Find the number.
number by 10. The
sides of a rectangle differ by 9 inches, and equals 190 square inches. Find the sides. 11.
A
its
area
rectangular field has an area of 8400 square feet and Find the dimensions of the field. feet.
a perimeter of 380
ELEMENTS OF ALGEBRA
190
The length
12.
AB of a rectangle, ABCD, exceeds its widtK AD by 119 feet, and the line BD joining
B 1
two opposite .
c equals 221
vertices (called "diagonal")
feet.
Find
AB and AD.
The diagonal
13.
of a rectangle is to the length of the recthe area of the figure is 96 square inches.
tangle as 5 4, and Find the sides of the rectangle. :
A man
14.
sold a
watch for $ 24, and lost as many per cent Find the cost of the watch.
as the watch cost dollars.
A man
15.
sold a watch for $ 21, and lost as many per cent Find the cost of the watch. dollars.
watch cost
as the
A man
16.
Two steamers
17.
of 420 miles. other,
and gained as many per Find the cost of the horse.
sold a horse for $144,
cent as the horse cost dollars.
and
is
ply between the same two ports, a distance One steamer travels half a mile faster than the two hours less on the journey. At what rates do
the steamers travel ? 18. If a train had traveled 10 miles an hour faster, it would have needed two hours less to travel 120 miles. Find the rate
of the train. 19. Two vessels, one of which sails two miles per hour faster than the other, start together on voyages of 1152 and 720 miles respectively, and the slower reaches its destination one day
before the other.
How many
miles per hour did the faster
vessel sail ?
If 20. A man bought a certain number of apples for $ 2.10. he had paid 2 ^ more for each apple, he would have received 12 apples less for the same money. What did he pay for each
apple ?
A man bought a certain number of horses for $1200. had paid $ 20 less for each horse, he would have received two horses more for the same money. What did he pay for 21.
If he
each horse ?
QUADRATIC EQUATIONS
191
--
On the prolongation of a line AC, 23 inches long, a point taken, so that the rectangle, constructed with and CB as sides, contains B 78 square inches. Find and CB. 22.
B
is
AB
AB
23.
A rectangular
24.
A
grass plot, 30 feet long and 20 feet wide, is surrounded by a walk of uniform width. If the area of the walk is equal to the area of the plot, how wide is the walk ? circular basin is surrounded
and the area of the path
is
-
by a path 5
feet wide,
Find
of the area of the basin.
=
the radius of the basin.
2
TT r (Area of a circle .) 25. A needs 8 days more than B to do a certain piece of work, and working together, the two men can do it in 3 days. In how many days can B do the work ?
26.
Find the side of an equilateral triangle whose altitude
equals 3 inches. 27. The number of eggs which can be bought for $ 1 is equal to the number of cents which 4 eggs cost. How many eggs can be bought for $ 1 ?
236.
EQUATIONS IN THE QUADRATIC FORM An equation is said to be in the quadratic form
if it
contains only two unknown terms, and the unknown factor of one of these terms is the square of the unknown factor of the other, as 0,
^-3^ = 7,
2
(tf- I) -4(aj*-l)
= 9.
237. Equations in the quadratic form can be solved by the methods used for quadratics.
Ex.
1.
^-9^ + 8 =
Solve
**
By formula,
Therefore
x
=
\/8
0.
=9
= 2,
or x
= \/l = 1.
ELEMENTS OF ALGEBEA
192 238. stitute
Ex.
In more complex examples it is advantageous to sub a letter for an expression involving a?.
+ 15 =
2.
x
,
or y
=
0,
8.
Le. Solving,
=
1,
EXERCISE Solve the following equations 1.
4 a; 4
-10a; 2 -h9:=0. 4-36
= 13.T
2.
a;
7.
3 a4
8.
16 a^-40
3.
2
4.
.
a4 -5o;2 =-4.
2
11.
4
6.
4
a -21or=100.
-44s + 121=0.
aV+9o
89
:
=0.
9.
10.
4
4
-8 = 2 a*
6.
-37aj 2 = -9. 2
2
(a:
4
+aj)
-18(x2 +a;)+72=0,
2 (^-Z) -
12.
"3 14.
1=2*. T
15
16.
^^
a?
17.
(a?-
18. 19.
^ 2:=Q>
~ 28
193
QUADRATIC EQUATIONS
CHARACTER OF THE ROOTS 239.
The quadratic equation
oa/*
2
bx
-f-
1.
2.
3.
it
follows 2
is
4c
is
a positive or equal to zero, the roots are real. negative, the roots are imaginary. a perfect square, the roots are rational.
4 ac
is
Iflr
kac
is 'not
4 ac
is zero,
4ac
is
2
(
:
4 ac
If b Ifb* 2 If b
Ifb 2 Jfb
has two roots,
2a
2a Hence
=
c
-f-
a perfect square, the roots are irrational. the roots are equal.
not zero, the roots are unequal.
240. The expression b 2 the equation ay? 4- bx 4- c
4 ac
is
called the discriminant of
= 0.
Ex. 1. Determine the character of the roots of the equation 3 a 2 - 2 z - f> = 0. The discriminant =(- 2) 2 4 3 (- 5) = 04. .
Hence the roots are
real, rational,
and unequal.
Ex. 2. Determine the character of the roots of the equation 4 x2 - 12 x + 9 = 0. 2
4
4
9
= 0,
the roots are real, rational, and equal.
Since
(
241.
Relations between roots and coefficients.
12)
the equation ax2 4- bx 4-
are denoted
c
__
b 4- Vfr 2
Tl
Vi
2
2a
Or
/ 1
4-r2
4 ac '
T* b
Hence
by
= a
,
4 ac
i\
If the roots of
and r2 then ,
ELEMENTS OF ALGEBRA
194
If the given equation is written in the form may be expressed as follows
these results
If the (a)
2 a?
+ a-x + -a =
0,
:
ofx
coefficient
2
in
a quadratic equation
The sum of the roots
is
equal
is
unity,
of x with
to the coefficient
the
sign changed. (b)
The product of the roots
is
2 E.g. the sain of the roots of 4 x
equal to theubsolute term, -f
5 x
3
=:
j, their
is
product
is-f.
EXERCISE
89 a
Determine without solution the character of the roots of the following equations
:
4.
= 0. 5a -26a? + 5 = 0. 2x* + 6x + 3 = 0. or + 10 + 4520 = 0.
5.
^-12.
6.
3a;2
7.
9x2 ~
1.
2
o;
-lla; + 18 2
2.
3.
a;
+ 4a: + 240 = 0.
2
+ 2-a;. = 5x. 12~x = x
8.
5aj
9.
x2 -7
10.
n
2
.
a?-3 '
12.
10 x
== ~
l
= 25 x + 1. 2
In each of the following equations determine by inspection sum and the product of the roots:
the
13.
14. 15.
= Q. -9a-3 = 0. 2a -4z-5 = 0. x2 -!i>x + 2
16.
z2
17.
2
18.
= 0. tfmx+p^Q. 5oj -aj + l = 0. Sa^ +
2
Ooj
2
Solve the following equations and check the answers by
forming the sum and the product of the roots 19.
20.
21.
a 2 - 19 #
= 0.
+ ^ + 2^-2 = 0. + 2a-15 = 0. 2
ar
60
:
22.
x2 -4 x
23.
0^
+
24.
or
j
-
205
= 0. = 0. + 12 2
CHAPTER XVI THE THEORY OF EXPONENTS 242. The following four fundamental laws for positive integral exponents have been developed in preceding chapters :
a m a" = a m+t1 . ~ a m -f- a" = a m n
I.
II.
m
mn . (a ) s=a m = aw bm
III.
IV.
The
first
,
provided
w > n.*
a
(ab)
of these laws
is
.
the direct consequence of the defiand third are consequences
nition of power, while the second of the first.
FRACTIONAL AND NEGATIVE EXPONENTS Fractional and negative exponents, such as 2*, 4~ 3 have meaning according to the original definition of power, and
243.
no
,
we may choose
for such
symbols any definition that
is
con-
venient for other work. It is, however, very important that all exponents should be governed by the same laws; hence, instead of giving a formal definition of fractional and negative exponents, we let these quantities be what they must be if the exponent law of multiplication is generally true.
244.
of
We assume,
m and n.
m therefore, that a
Then the law
*The symbol
>
an
= a m+n
of involution, (a m ) w
means "is greater than"
smaller than."
195
;
,
for all values
= a""
similarly
i""i
ELEMENTS OF ALGEBRA
200
Solve the equations
:
= l. ar = i. 2 =f 3* = f x~
46.
l
2
47.
z
48. 49.
Find the values
= 1. = 5.
= .1. = -^.
50.
17'
54.
10*
51.
z*
55.
5*
52.
5or*=10.
53.
10*
= .001.
of:
56. 57.
3-ll-
58.
4~*
59.
60.
61.
+ 1~* -f 21 - 9*. (81)* + (3f)*-(5 TV)*-3249 + 16 * - 81 -f (a - 6). - (.008)* + A. + A_. (.343)* + (.26)* 1
(I-)
2
.
5
-
75
USE OF NEGATIVE AND FRACTIONAL EXPONENTS 249. It can be demonstrated that the last three laws for any exponents are consequences of the first law, and we shall hence assume that all four laws are generally true. It then follows
that:
Fractional and negative exponents
may
be treated by the
same
methods as positive integral exponents. 250.
Examples relating
to roots can be reduced to
taining fractional exponents.
Ex.
1.
Ex 2
(a*&~*)*
+
(aVM = a*&~* +
V
'
=
'*&*
examples con-
THE THEORY OF EXPONENTS
201
Expressions containing radicals should be simplified as
251.
follows
:
(a)
Write
(6)
Perform the operation indicated.
(c)
Remove
all radical
signs as fractional exponents.
the negative exponents.
remove the fractional exponents.
(d) If required,
NOTE.
Negative exponents should not be removed until all operations of multiplication^ division, etc., are performed.
EXERCISE Simplify
2.
&.&.&.$-".$-*.
3.
72
4
25
26
5.
a- 3
-
6.
aj"
7.
6a-.5a.
8
6 *- 6 *' 6 *-
'
.
92
:
79
.
7~ 5
-
3 a- 4
a8
OA 20.
.
2~ 9
2~ 8
27
a- 4
7~ 6
.
.
2 a?
2 ar 1
.
22.
3-s-VS.
23.
/ 7-f--v 7.
25. 9.
7*.7i.7*.7W. .
26.
,
4 x^.
10.
#*
11.
V5.^/5-^5.
27
12.
95 -^9i
28.
13.
5-*-*- 5.
14.
S-'-s-S-8.
16.
a9 -i-a- 4
16.
14an-
17.
(4**-
18.
(Va)
'
a;
4 .
.
-
__ 29-
/m '-=V--
ELEMENTS OF ALGEBRA
202
V ra 4/
32.
-\/m
3
6
33. 34.
35.
40.
we wish to arrange terms according to descending we have to remember that, the term which does not contain x may be considered as a term containing #. The 252.
If
powers of
a?,
powers of x arranged are
Ex.
1.
:
1 Multiply 3 or
+x
Arrange in descending powers of
5 by 2 x
1.
x.
Check.
lix
=+1
2x-l
Ex.
2.
=
Divide
by
^ 2a
3 qfo
4- 2 d
THE THEORY OF EXPONENTS EXERCISE
203
93
Perform the operations indicated:
2. 3.
4. 5.
(7r-8Vr + r>)(9 Vr-7). 2 - 1 ). (a- + a -f 1) (a~ + a 2
2
2
6.
7. 8. 9.
10. 11.
12.
13.
14.
(4
a- 3
- 24 a- - 9 - 3 a~ )
1
2
1
-r-
(a"
- 3).
+ + 47i) + 35V5?)-*-(5Vp + l). VS" ^- ( Vo Vft) 1C H- (a~ -f 7 a- ^a~ a-*b~ 33 a6~ + 14 a+ (3 a _&)-*. (-^? + ^/-^ + */fr^ 15. 16. (a-6 + 2V6c c)-^-(Va+V6 Vc). 17. -y^TTOa; -f 13 - 12 *- + 4 aF*. (13Vp
l
(Va^-f aV^-&Va 5
3
)
3
l
2
2
^>~
2
3
1
1
)
(
1
18.
Vor
19.
V25 #
20.
^^
21. 22. 23. 24. 25.
l
2
2 x -h or 2
2 or
1
-f-
3.
- 2()"ar r+ 34 - 12 x -f 9 x*.
+2
(l+4^-flO^ + 20oT-f 25^T -f-24-\/i?-f 16 (1+V2)V2. (2+V2)(V2-2). (5+V3)(5-2V3).
a?
8
)*.
26.
(1-3VS)(2 + V5).
27.
(VU - V2)(Vn~3V2)
ELEMENTS OF ALGEBRA
204
Find by inspection 28. 29.
:
(x*
+ 3)(tf*-f 2).
a*
+ 3l-5.
35. 36.
8 (a;*
yi)
(5*-2*
.
2 .
30. 31.
V-
38.
2
32.
(3^
33.
(#* ^
2*)
.
39.
34.
(fl
-f-
5) (x*
5).
40.
(m
n)
-f-
11
(m*
-f-
n 5 ).
CHAPTER XVII RADICALS 253.
A
a quantity, indicated by a
radical is the root of
radical sign.
254.
The
radical is rational, if the root can be extracted
exactly; irrational, if the root cannot be exactly obtained. Irrational quantities are frequently called surds.
^9 4^
255.
The
+ V) *
are radicals.
= 2, V(a + 6) 2 are rational.
\/2,
root.
(*
V4a-f
b are irrational.
order of a surd /-
is
indicated by the index of the
va
is
of the second order, or quadratic.
\/2
is
of the third order, or cubic.
Vc
is
of the fourth order, or biquadratic.
.
256. A mixed surd is the product of a rational factor and a surd factor; as 3Va, a;V3. The rational factor of a mixed surd is called the coefficient of the surd.
An 257. factor.
entire surd is
one whose coefficient
Similar surds are surds 3v/2 and 6
3V2 and
is
unity; as
Va,
which contain the same irrational
av^
are similar.
3 V8 are dissimilar.
206
ELEMENTS OF ALGEBRA
206 258.
Conventional restriction of the signs of roots.
All even roots
may
be positive or negative,
VI = + 2
e.g.
or
2.
Hence 6. which results in four values, viz. 14, 6, To avoid 14, or this ambiguity, it is customary in elementary algebra to restrict
the sign of a root to the prefixed sign.
5 V4 4- 2 V4
Thus
= 7 VI = 14.
If the object of an example, however, is merely an evolution, the complete answer is usually given thus ;
=-
(oj- 2).
Since radicals can be written as powers with fractional
259.
exponents, all examines relating to radicals
may
be solved by the
methods employed for fractional exponents.
Thus, to find the nth root of a product ab we have T
1
1
(a6)"==a"6" I.e.
(242).
to extract the root of a product, multiply the roots of the
factors.
TRANSFORMATION OF RADICALS 260.
Simplification of surds.
A radical is simplified when the
expression under the radical sign is integral, and contains no factor whose power is equal to the index.
Ex.
1.
Simplify
= \/25~a~ Vb = 6 a*VS. 4
Ex.
2.
Simplify
-v/16.
-J/lB^^.
4/2
= 2^.
RADICALS
207
261 When the quantity under the radical sign is a fraction, we multiply both numerator and denominator by such a quantity as will make the denominator a perfect power of the same .
degree as the surd.
Ex.
3.
Simplify V|.
Ex.
4.
Simplify
EXERCISE
94
ELEMENTS OF ALGEBRA
208
/s
39.
37.
j
*x+y 38.
262.
An
\ 2m
n
imaginary surd can be simplified in precisely the as a real surd thus,
same manner
;
,
42.
V-16a
2
44.
.
2\-
Simplify and find to three decimal places the numerical values of 47. 48.
:
VJ.*
49.
Vf.
VJ.
50.
VA
263.
Reduction of a surd to an entire surd.
Ex.
Express 4 a V& as an entire surd.
EXERCISE Express as entire surds
95
:
1.
4V5.
3.
2-\/lL
5.
2.
3V7.
4.
3^5.
6.
7.
a VS.
8.
* See table of square roots on page 164.
RADICALS
209
264. Transformation of surds to surds of different order.
Ex.
1.
Transform -\/uW into a surd of the 20th order.
Ex.
2.
Transform
lowest order.
\/2,
V3, and
\/5 into surds of the
same
V2 = 2* = a* = '#64. |^ = 8* = 3A= ^gi. ^5 = 6* = 6* =^125. 1
Ex.
Reduce the order of the surd tyaP.
3.
Exponent and index bear the same relation as numerator and denominator of a fraction ; and hence both may be multiplied by
same number, or both divided by the same number, without changing the value of the radical. the
EXERCISE Reduce 1.
Va?.
96
to surds of the 6th order
-fymn.
2.
Reduce
3.
\/ v
:
4.
v'c?.
to surds of the 12th order
7.
V2~a.
8.
^v/mV
9.
10.
5.
\|
z
\
^-
6.
mn.
3
:
\/a4 6 2c.
11.
-\/oP6.
13.
-\/3ax.
12.
\/5a5V.
14.
a.
Express as surds of lowest order with integral exponents and indices :
5
15.
-v/o
20.
A/^
.
16.
\/oW.
22.
17.
VSlmV.
2
-v/IaT .
24.
18.
-\/
ELEMENTS OF ALGEBRA
210
Express as surds of the same lowest order
:
32.
25.
V3,
1
lla
7.
8-
10
12~
13. 9.
10.
III.
HOMOGENEOUS EQUATIONS
A
homogeneous equation is an equation all of whose terms are of the same degree with respect to the unknown 295.
quantities. 4^ 3 x 2 y
3 y3
and # 2
2 xy
5 y2
are
homogeneous equations.
one equation of two simultaneous quadratics is homogeneous, the example can always be reduced to an example If
296.
of the preceding type. '
Ex.
1.
Solve .
Factor (2),
x*- 3 2x
(x
Hence we have
to solve the
2
y*
+ 2y = 3,
7 xy
(1)
+ G if = 0.
2t/)(2 x
3y)
two systems
=
(2)
(
:
(3) (1)
From
x-2y.
(3),
Substituting in (1),
4 f-
Hence
3 y2
y
=1
+ 2 y = 3,
8
y ,
1
3 3,
':il -e :)
=
V-~80
ELEMENTS OF ALGEBRA
236
297. If both equations are homogeneous with exception oi the absolute terra, the problem can be reduced to the preceding case by eliminating the absolute term. =
Ex. 2
Solve
.
(1)
2,
Eliminate 2 and 6 by subtraction. (1)
x
5,
(2)
x
2,
15 x2
15 y 2
= 2 x 5.
(3)
(4)
= 0. = (rc-2/)(llx-5y) 0.
11 a2
Subtracting,
Factoring,
Hence
- 20 xy +
solve
16 xy -f 5 y 2
:
(3)
(2)
From
(3),
j
Substituting y in (2),
109
^ EXERCISE
a;2
VI09, y
=
110
Solve: 6ar --7aK/4-27/2 ==0, }
f
10^-370^ + 7^ =
16^-7^
SIMULTANEOUS QUADRATIC EQUATIONS
m
.
f^ +
3 7/
= 133,
= 189, '
*
Some simultaneous
B.
300.
considering not x or 2
-, xy,
x
x
,
x
+y
y
etc., at first
more complex examples letter for
Ex.
1.
quadratics can
be solved by
but expressions involving x and
?/,
it is
as the
unknown
quantities.
?/,
as
In
advisable to substitute another
such expressions. Solve
i"
1
+
1
4
J,
-J,
ELEMENTS OF ALGEBRA
254 Ex.
Find the value of .3727272
2.
.;)7?7272
The terms afteAhe
=
...
first
.3
+
form an a
+
.072
= .072, 1
Therefore
.37272
-
=A+
. . .
10
1.
2.
1,
1,
i i
J,
1,
.99
i. 65
= 1L
9, 6, 4, ....
If
a
9.
10.
15. is J.
16. 8.
4.
= 40, r = j.
Find the value
= .72. = 990
.
66
.
110
118
16, 12, 9, -..
3.
....
8.
^
.01
infinity of the following series
-.
7.
....
= .Ql.
r
EXERCISE Find the sum to
.00072 -f
infinite G. P.
.= _4Z* - =
Hence
....
3,
- 1,
i,
-.
Find the sum to
:
5.
5, 1, I,
6.
250, 100, 40,
....
infinity.
of:
.555....
11.
.191919-...
13.
.27777
.717171-...
12.
.272727-..
14.
.3121212-..
The sum
of an infinite G. P.
Find the
first
The sum
of an infinite G. P.
Find
...
is 9,
....
and the common
ratio
term. is 16,
and the
first
term
is
r.
17. Given an infinite series of squares, the diagonal of each equal to the side of the preceding one. If the side of the first square is 2 inches, what is (a) the sum of the areas, (6) the sum
of the perimeters, of all squares ?
BINOMIAL THEOREM EXERCISE
Expand
2.
the following
(x-y)
6 .
119
:
+ xy.
3.
(1
4.
(a-2)
7 .
5.
(s
+ i).
7.
6.
/2a+|Y-
8.
9.
l
(z2
4
\
Simplify
257
:
4 (1+V#) + (1
Va)
4 .
10.
(\
+ b) w (a b)
9
11.
Find the 5th term of
12.
Find the 3d term of
13.
Find the 4th term of (w
(a
.
.
12
-f
.
ri)
+ a)
11
14.
Find the 5th term of
15.
Find the 4th term of
16.
Find the 6th term of (x - a2) 25
17.
Find the 5th term of
18.
Find the 3d term of fa -f
(1
7 (a -f 2 b) .
f
Find the
20.
Find the
.
Vx + -^r
-^Y
-
Va/
V 19.
.
u 13 coefficient of a?b in (a -f 5) .
21.
a4 b 12 in (a -f 6)16 Find the coefficient of a5 b 15 in (a - 6) 20
22.
Find the coefficient of a?V" in (a
coefficient of
.
8
16
100
.
2
2
23.
Find the
24. 25.
Find the middle term of (x + y) 4 Find the middle term of (a b)\
26.
Find the middle term of
27.
Find the middle term of (m ri) 16 Find the 99th term of (a + b) m im Find the 1000th term of
coefficient of
a6
in
.
f f
x
}\8 :
)
28. 29.
.
.
(a
.
- 6) - b ). (a
+ b)
.
-^
ELEMENTS OF ALGEBRA
258
REVIEW EXERCISE Find the numerical values 1.
~
27 x*
27 x-y
*=M
or
y=2j 2.
-
16 x*
9 xy~
-f
32 afy
4 *2
-
4 xy
-
-
13 a a b
1
]
]
1
1
1
lj
2j
3}
4j
2J
4J
a 8~T +
a2
+
3 2 ft'
ft
c
8
y
,
3,
4.
=
3,
2,
3,
4,
3,
4,
5.
2,
^+ = = =
2,
3,
3,
3,
4,
4,
5.
4,
2,
3,
4,
3,
2.
2,
1,
1,
2,
2,
1,
3,
6.
38
aft
+
24
= =
3
ft
-
4
)
2 (2 a
2,
2,
3,
3,
4,
4,
5,
6.
1,
2,
1,
2,
2,
4,
2,
3.
5.
a 2^
+
3 a l} 2
-r
ac
aft
-
-
3
aft
2,
4,
4,
5,
5,
5,
2,
1,
5,
1,
3,
5,
6.
3,
3,
2,
1,
2,
4,
2,
7.
+
a)(a
3,
4,
1,
2,
2,
1,
2,
2,
1,
l,
2,
1,
3,
3,
ft)
- a(a
4-
4,
2,
2.
1,
3,
3.
ft
-
c)
+
c(a
-3, -5, -6. c
7 a 6
c
= = =
2,
-2,
1,
1,
3,
-3,
if
i
,
(a-ft)(a-c)
(ft- c )(ft-a)
-
-f-
4
2 ft
), if
,f
1,
-
-
be
2,
(c
5J
if
,
2,
31 a 2 ft 2
3,
2
1,
a
)
^+^
2
2,
+
= =
+
2 ?/
M.
if
3,
4
= = =
+ 2,
c
a
8
2,
;]
4-
^
2,
3 -r C T + + + c2 + 2
c)(c
8
3
1,
ft
(ft
-
3
2
2
x^l,
a
6.
2
24 afya
-f
ft
5.
if
,
M
?/
4 (2 a
#
3J
a:
4.
:
8
2
y 3.
of
-
(c-a)(c-ft)'
-4 -
1,
-1,
3,
4,
2,
-
2,
2,
4,
-3, -1, -1, -3,
2,
2,
+
1.
1.
2.
-|-
c), if
259 x
c)
(b
6-)
= 1, = 2, c = 3, x = 4, a
1,
/>
9. a, by
The
and
Find
Add
2, 5,
(c
+
(5,
g)(x
c(x
,
-
-
a) (c
3,
2,
4,
6,
5,
1,
2.
2,
3,
1.
-
'
b)
2.
radius r of a circle inscribed in a triangle whose sides are by the formula
= = c =
a
3,
10,
8,
6
4,
21,
17,
24,
5,
26,
15,
7,
r, if
41.
29,
25,
9.
21, 20,
40.
the following expressions and check the answers
-
11. x 2
+ -
x
4
2 ax*
y
+
4
-f
a zx
z8
-
3
4x y
xy -
C
+
+
6 x
a8
4 x2
,'
12 xy*
2
+
2 ?/
x3
,
-
-
5 z3
4 xy
G y4
,
8
,
+
8
4 xy*
2
.c'
-
4 ?y
-
,
4
-
2 a3
3 ax'2 ,
-f
7 y4
4 x 4 /
,
-f
-
3 //
y
ax'2
-
10 z 8
2
x'
:
2 x 8. 6 y4
,
zy +
12
-
6 2 8.
12 xy*
-
4 y4
4 .
+ x/y 2 + + y'2z + 2 3 x 10 y'2 + 5 z2 - 7 ys, - x 2 + 4 2 ~ 10 z 2 + z 2 + 11 yz + 8 2:2 - 2 x?/, 4 z - \ yz + xz, 2 2 x2 + and 9 2:2 y' xy. 1 + 3 x + 2 x 8 - x 5 4 - 2 x2 - 8 3 + 7 x4 - 4 x'2 -f 12 x and 5 2 + 7 x8 - 11 x 5 12
13. x 3
14.
3,
a}
~c)(b- a) - 1, - 2, 3, + 1, -f 8, 4, - 2, - 4, 5, + 2, + 4,
c is represented
10. x 3
12.
c)(x
b(x
.
-
.r
z
3
7/
ary,
2
15.
16.
11 z 4
-
-12
x4
-
-
2
9
11 x 8
.
4 2 */, 7 xy 3 - 2 a?y + 3 aty - 8 y y 3 4 8 5 3 5 4 * + + xy a?y y, 7y 4
,
*y
+ 12 a 8 - 10, a 4 + 11 a - a 5 ,
14
6 a4
4 a8
-7xy* +
x^ij
+
8 x4
.
a
+
,
or
,
17. 4 a 5
18.
a:
,
r>
a;
- a8 - 7 + - a 4 - 5.
z 3,
x3
-
2 x 2//
2 a2
+
3
,
4a
2 x?/
+
-
7 y3
+ 3 y 2* - 2 z8 4 x- 8 + 2 // - 11 z 3 4 4 ?p 2 - 3 xyz, and 3 y 8 -f 12 z 8 - 7 y 2* 4- 4 xyz + 4 xy'2 - 4 yz\ ,
9 a2
-
3 a5
,
,
ELEMENTS OF ALGEBRA
260
4-X-5V14- #4-8, - 4\/i + x 3Vl 4- x 4- 4 Vl 4- 4- 3, and 2 Vl 4-
19.
6 VI
20.
Take the sum
2VT+7 - ?> x
*/
6 x8
4x
4-
-2
a 2x
From
G x 4y
2 y5
-x
2
4-
the
2 x2
4-
2
xy
From
sum
26.
of 2
From sum of 2
-
29.
1
from
Add
0" 30.
find
(a) a ft
Simplify
34.
12 x 5
4-
2 xs
4
take the
4
- 2 x 8y2 44 - x - x2
of
of
-
2 c
the
-
ft
G x2
sum
G x
3 a,
4- c 4-
ft
2 c
a,
of
-
2 c
5 10 ,
6
4-
c 4-
2 a
4-
3
4- c
ft
-
2 c
a,
- 5 10 -
x4-y4-2, -f
-
ax 2
and
a'2x,
2
4-
4-
x2
2 x6
,
7 x
x'2 . 4
4 x 4 ?/
/-
2 x2
-f
and 4
4- 5,
3
4-
5 y/
-2x
x5
,
3
4-
x// 5 ?/
4
,
#
.
-
4
-
5
54-2 x 2 and
x*,
,
7 x
b
=
7 12
.
x
c,
(