Algebraic Equations of Arbitrary Degrees (Little Mathematics Library)

A. I Kypour AJII

ESPAMtIECKHE

IIPON3BOJlbHbIX H3gRTe.1bCTBO «Hayrca»

YPABHEHNSI

CTEl1EHEA

A.U.

Kurosh

ALGE

B RAIC

EQUATIONS OF ARBITRARY

DEGREES Translated

from

by V.

Kisin

the Russian

t erst punhsned I/li Revised from the l975 Russian edition

HQ OHZABQCKO.u A3slKC ! English

translation, Mir Publishers, 1977

t 'ontents

Preface

7

troduction

9

Complex Numbers

10

In 1.

Evolution. Quadratic Equations

te

2.

Cubic Equations

19

3.

Solution of Equations in Terms of Radicals and the Existence of Roots of Equations

22

The Number of Real Roots

24

Approximate Solution of Equations

27

6.

Fields

30

7.

Conclusion

35

Bibliography

36

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Prefact

the author's lecture to high Mathematics Olympiad at a review

of the results

and

algebraic equations with due >f its readers. No proofs are have required copying almost ,her algebra. Despite such an eke for light reading. Even a ir the reader's concentration, initions and statements, check , application of the methods

This

booklet

is a revision

of

school students taking part in thi Moscow State University. It gives methods of the general theory of regard for the level of knowledge i included

in the text since this would

half of a university textbook on hi~ approach, this booklet does not m: popular mathematics book calls fc thorough consideration of all the del of calculations in all the exatnples described to his own examples, etc.

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Introduction

A secondary school course of algebra is diversified but equations its focus. Let us restrict ourselves to equations with one tnown, and recall what is taught in secondary. school. Any pupil can solve jirst degree equations: if an equation ax+

are unl

b =0

~Jven,in which a 4 0, then its single root is X=

is

a

Furthermore, a pupil knows the formula l'or solving quadratic survives. Hoi

1 =0

t of such equations. Is there a way to expand the tbers so that these equations also possess roots? !dy of mathematics at school the student sees numbers at his disposal constantly extended. He .grai positive numbers in elementary arithmetic. Very .........1.

The theory theory of co:

s.ne. gati.ve..~umber'...a.i.

J

is the simples realm In

of nun his st
0, then the positive value of one

:gative one to the

to the positive value of the' other, and the ne

~ to the following radical

ar equation are the numbers : 3+i,

xg

1

2i

d that

each

of

these

Therefore, the roots of oi xg =

numbers

indeed

the problem of extracting roots of an index n from complex numbers. It can :omplex number m there exist exactly n

> such that raised to the power n i. e. n factors equal to this number!, each other words, the following I

extremely

It can easily be checke satisfies the equation. I.et

to

theorem

holds.

A root of order n distinct

y applicable to real numbers, which are tplex numbers: the nth root of a real distinct values which in a general case that among these values 'there will be ers, depending on the sign of the number

turn

arbitrary positive integral be proved that for any < distinct complex number< if we take a product of yields the number a. In itn ortant

>f any complex number has exactly n

< n. f one

us now

cont lex values.

This t eorem is equaH

a particular case of con number a has precisely n are complex. We know two, one or no real numb

a and parity of the inde> has three values:

l and 22

i

Thus, the cube root

y~

1,

2

the formula for taking the square root + bi, This

formula

reduces

calculation

quare roots of two positive real numbers. no formula exists which would express

1 2

c

+

In section 1 we gave

of a complex number a of the root to extr'actinge @Unfortunately, for n o 2

bi in tertns of real values

the nth root of a compie

iliary real numbers; it was proved that be:derived. Roots of order n of cotnplex

of radicals of certain aux no such formula can ever

x number

a+

Cubic Equations

g3.

ig quadratic equations is also valid for iird-degree equations, usually called cubic erive a formula, which, although more h radicals the roots of these equations 'his formula is also valid for equations

for solviti

complex coefficients. For th equations, we can also d complicated, expresses wit in terms

of coefficients.

'I

with arbitrary complex co Let an equation

efficients.

- ax

The formula

+ bx + c = 0

x

iis equation, setting

be given. We transform tl

a

x=y

3

a. Substituting this expression of x into cubic equation with respect to y, which

cient of y~ will be zero. The coefficient l the absolute term will be, respectively,

where y is a new unknowi our equation, we obtain a is simpler, since the coeffi of the lirst power of y ani the numbers

tb> written '+

il=

2a 27

ab 3

+c i. e. the equation can be

as + =0

+ p + ' 27+ the three cube radicals

has three values.

iot be combined in an arbitrary manner.

We know that each of

However, these values cani

se two values of the radicals

the number

must be added

root of the equation. Thus we obtain the uation. Therefore, each cubic equation with has three roots, which in a general case are mmeof these roots may coincide, i, e, constitute

P 3 The

together to obtain a three roots of our eq numerical

coefficients

complex; obviously sc a multiple root, The practical sigr small. Indeed, let th
x" ' +

+a

Let an nth

ix + a= 0

efficients, We already know that it has n roots. m real roots? If so, how many and approximately located? We can answer these questions as follows. he polynoinial on the left-hand side of our equation

having real cc Are any of the where are they Let us denote t

by f x!, i. e. x! = aox" + a,x"

++

a,x+

a

iiliar with the concept of function will understand he left-hand side of the equation as a function of Taking for x an arbitrary numerical value m and

The reader fan that we treat t the variable x.

into the expressionfor f x!, after performing all the

substituting it i operations, we of the polynon

arrive at a certain

iial f x!

number

which is called the

value

and is denoted as f cx!. Thus, if

f x! = x' Sx'

+ 2x + 1 and m=2,

f!

=2

5

2 + 2. 2+ 1 = 7

: a graph of the polynomial f x!, To do this we ordinateaxes on the plane see above! and, having i value

the

Ix and calculated

s I I I I I]I I

a corres

II I I I I I I I » ]IqlI I I I I i

ondin

value

m

Let us plol choose the coc selected for x

I ]II I I I I I

question. :ly, since there are an infinite number of the values

>t hope to lind the points n, f u!! for all of them satisfied with a finite number of points. For the ity we can first select several positive and negative of »x in succession, mark on the plane the points to them and then draw through them as smooth

Unfortunat» of »x one cann» and must be:

sake of simplic integral values corresponding

and [a~ =

a for a < 0! and A is the greatest of the absolute

values of all the other coefficients a,, a2, B=

A

I ~0I

acr,

then

+1

However,it is often apparent that these bounds are too wide. Example. Plot a graph of the polynomial

f x! = x

5x

m 2x + 1

Here ao! = 1, A = 5, and thus B = 6. Actually, for this particular example we can restrict ourselves to only those values of m, which fall between 1 and 5. Let us compile a table of values of the

polynomial f x! and plot a graph Fig. 3!.

The graph demonstrates that all the three roots a,, e, and a3

i!

li 1I

i!

1 I!

1

I

neighbouringvaluesof a for which the numbersf m! have opposite signs, and thus it was sufficient just to look

ht the table of

values of f a!. If in our example we found less than three points of intersection

of the graph with the abscissaaxis, we might think that owing to the imperfection of our graph we traced the curve knowing only seven of its points!, we could overlook several additional

cated between any given

equation and even the number of roots lo

thods will

numbers here.

not be stated

a and b, where a

b, These me

Sometimes the following theorems are useful since they give some information

on the existence

of real

and even positive roots. Any equation of an odd degree with real coefftcients has at least one real root.

If the leading. coefftcient ao and the absolute term a in an equation with real coefftcients have opposite signs, the equation has at least one positive root. In addition, if our equation is of an even degree,it also has at least one negative root.

Thus, the equation x'

8x

+x

2=0

has at least one positive root, while .the equation -x +2x'

se neighbouring integers ,tion

x

1=0

In the previous section we found tho, between which the real roots of the equa

I0

e roots of this equation :xample, let us take the

+7x

x are located.

The

5x

same method

+2x+ allows

I= th

to be found with greater accuracy. For < If1 4

sive values >scissa axis, f one-tenth. root u2 to

0.9, we can find between which two of these succes

of x the graph of the polynomial f x! intersectsthe af

methods of :ions much :thods and of the Ful to find

i. e. we can now calculate the root u2 to the accuracy o Proceeding further, we can find the value of the the accuracy of one-hundredth, one-thousandth or, tl to any accuracy we want. However, this approac cumbersome calculations which soon becoine practica ageable. This has led to the development of various . calculating approximate values of real roots of equai quicker. Below we present the simplest of these mi immediately apply it to the calculation of the root cubic equation considered above. But first it is use

>ady know,

bounds

oot to the lues of the

0 < m, < 1. For this purpose we shall calculate our r accuracy of one-tenth. If the reader calculates the val polynomial

ieoretically, h involves

lly unman-

for this root

narrower

f x! = x for x = 0.1; 0.2;

'0.9, he will

f.7! re difFerent,

than

the ones we aln

5x + 2x + 1 obtain

= 0.293, f.8!

=

0.088

and therefore,since the signs of these values of f x! a: 0.7 < >x, < 0.8

n is given,

The method is as follows. An equation of degree

The bound c is calculated by means of the formul,

bf a! af ! f a! - f b! i! and f b!

In this case a =0.7, b = 0.8, and the values of f i are given above. Therefore

iction of a new

The formula for the bound d requires the introdv

:re; in essence

concept which will play only an auxiliary role h
f which may be multiple!. Even

equation will have n roots some