Higher Algebra [First Edition, Reprint]

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Higher Algebra [First Edition, Reprint]

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DAMAGE BOOK

DO

_1 60295

^

DO

>

HIGHER ALGEBRA BY S.

BARNARD, M.A.

FORMERLY ASSISTANT MASTER AT RUGBY SCHOOL, LATE FELLOW AND LECTURER AT EMMANUEL COLLEGE, CAMBRIDGE

AND J.

M. CHILD,

B.A., B.Sc.

FORMERLY LECTURER IN MATHEMATICS IN THE UNIVERSITY OF' MANCHESTER LATE HEAD OF MATHEMATICAL DEPARTMENT, TECHNICAL COLLEGE, DERBY FORMERLY SCHOLAR AT JESUS COLLEGE, CAMBRIDGE

LON-DON

MACMILLAN

fcf'CO

LTD

*v

NEW YORK

ST MARTIN *S PRESS

1959

This book

is copyright in all countries which are signatories to the Berne Convention

First Edition 1936

Reprinted 1947, ^949> I952> *955, 1959

MACMILLAN AND COMPANY LIMITED London Bombay Calcutta Madras Melbourne

THE MACMILLAN COMPANY OF CANADA LIMITED Toronto ST MARTIN'S PRESS INC

New York

PRINTED IN GREAT BRITAIN BY

LOWE AND BRYDONE (PRINTERS) LIMITED, LONDON, N.W.IO

CONTENTS

ix

IjHAPTER

EXEKCISE

XV

(128).

Minors, Expansion in Terms of Second Minors (132, 133). Product of Two Iteterminants (134). Rectangular Arrays (135). Reciprocal Two Methods of Expansion (136, 137). Use of Double Deteyrrtlilnts,

Symmetric and Skew-symmetric Determinants, Pfaffian (138-

Suffix,

143),

ExERtad XVI

(143)

X. SYSTEMS OF EQUATIONS. Systems (149, 150). Linear Equations in Line at Infinity (150-152). Linear Equations in Three Unknowns, Equation to a Plane, Plane at Infinity (153-157). Definitions, Equivalent

Two Unknowns, EXEKCISE XVII

(158).

Systems of Equations of any Degree, Methods of Solution for Special Types (160-164).

EXERCISE XVIII

XL

(164).

RECIPROCAL AND BINOMIAL EQUATIONS. The Equation Reduction of Reciprocal Equations (168-170). xn - 1=0, Special Roots (170, 171). The Equation xn -A =0 (172). The Equation a 17 - 1 ==0, Regular 17-sided Polygon (173-176). EXERCISE

XIX

(177).

AND BIQUADRATIC EQUATIONS. The Cubic Equation

(roots a, jS, y), Equation whose Roots are Value of J, Character of Roots (179, 180). Cardan's Solution, Trigonometrical Solution, the Functions a -f eo/? -f-\>V> a-f a> 2 4- a>y (180, 181). Cubic as Sum of Two Cubes, the Hessftfh (182, 183). Tschirnhausen's Transformation (186). (

-y)

2

,

etc.,

EXERCISE XX (184). The Biquadratic Equation

A=y + aS,

(roots a,

,

y, 8) (186).

The Functions

the Functions /, J, J, Reducing Cubic, Character of Roots (187-189). Ferrari's Solution and Deductions (189-191). Descartes' Solution (191). Conditions for Four Real Roots (192-ty). Transformation into Reciprocal Form (194). Tschirnhausen's Transetc.,

formation (195).

EXERCISE

XXI

(197).

OP IRRATIONALS. Sections of the System of Rationals, Dedekind's Definition (200, 201). Equality and Inequality (202). Use of Sequences in defining

a Real Number, Endless Decimals (203, 204). The Fundamental Operations of Arithmetic, Powers, Roots and Surds (204-209). Irrational Indices, Logarithms (209, 210). Definitions, Interval, Steadily Increasing Functions (210). Sections of the System of^Real Numbers, the Continuum (211, 212). Ratio and Proportion, Euclid's Definition (212, 213).

EXERCISE XXII

(214).

CONTENTS

x CHAPTER

XIV/INEQUALITIES. Elementary Methods (210, 217)

Weierstrass' Inequalities (216).

For n Numbers a l9 a 2 a

n (a* -!)/*

(a" -I)/*,,

xa x ~ l (a-b)$a x -b x

(l+x)

n

^l+nx,

\*JACJJ

>

n

n

(219).

^ xb x

~l

(a

- 6),

(219).

(220).

Arithmetic and Geometric Means (221, 222). -

-

^

V

n and Extension

(223, 224).

(224).

XV. SEQUENCES AND Definitions,

Maxima and Minima

(223).

EXERCISE XXIII

LIMITS.

Monotone

Theorems,

Sequences

(228-232).

E*

ponential Inequalities and Limits, /

l\m

/

i\n

and

1) >(!+-) m/ n/ \ /

1 \

n

lim (1-fnj n_ >00 V

EXERCISE

XXIV

l\-m /

/

=e,

nj

(1)

if

m>n,

(232,233).

(233).

General Principle of Convergence (235-237). Limits of Inde termination (237-240).

Theorems

~n

l\"w

=lim(l--) \

1 \

oo

lim (a n 6 1 n >ao

fe

w

= 6,

then

n~>oo

+ a w _ 1 6 1 + ...H-a 1 6 n )/n=o6,

(240-243).

Complex Sequences, General Principle of Convergence EXERCISE

XXV

I.

n->oo

>oo

(244).

(243, 244).

CONTENTS XVI. \CONVERGENCE OP SERIES

xi

(1).

Definitions, Elementary Theorems, Geometric Series (247, 248). Introduction and Removal of Brackets, Series of Positive Terms. p D'Alembert's of Order Terms, Comparison Tests, 271 /w Changing ,

and Cauchy's Tests

XXVI

EXERCISE

(248-254).

(254).

Terms alternately Positive and Negative (256). Absolute Convergence, with Terms Positive or Negative.

Series with Series

General Condition for Convergence, Pringsheim's Theorem, Introduction and Removal of Brackets, Rearrangement of Terms, Approximate Sum, Rapidity of Convergence or Divergence (256-261). Series of Complex Terms. Condition of Convergence, Absolute Conn n vergence, Geometric Series, Zr cos nd, Sr sin n6. If u n /u n+l = l+a n /n, where a n ->a>0, then u n -*Q. Convergence of Binomial Series (261-263).

XXVII

EXERCISE

(264).

^

XVII^CoNTiNuous VARIABLE.

Theorems Meaning of Continuous Variation, Limit, Tending to on Limits and Polynomials (266-268) Continuous and Discontinuous Functions (269, 270). Continuity of Sums, Products, etc., Function Fundaof a Function, lirn {f(x)}, Rational Functions, xn (271). ,

.

mental Theorems

Derivatives, Tangent to a Curve, Notation (272). of the Calculus, Rules of Differentiation (273-277). Continuity of of {f(x)} and xn (278, 279). Meaning of Sign {f(x)}, Derivatives of J'(x) (279). Complex Functions, Higher Derivatives (2*9, 280). Maxima and Minima, Points of Inflexion (280-282).

EXERCISE XXVIII

(282).

Inverse Functions, Bounds of a Function, Rolle's Theorem, MeanValue Theorem (284-288). Integration (289). Taylor's Theorem, Lagrange's Form of Remainder (290, 291). Function of a Complex Variable, Continuity (291, 292).

XXIX

EXEBCISE

(293).

XVIIL, THEORY OF EQUATIONS TIONS

(2),

POLYNOMIALS

RATIONAL FRAC-

(2),

(1).

Multiple Roots, Rolle's Theorem, Position of Real Roots of/(&)=0 Newton's Theorem on Sums of Powers of the Roots of f(x) =0 (297). Order and Weight of Symmetric Functions (298, 299). (296, 296).

Partial Derivatives, Taylor's x, y,

...

Euler's

.

(299-302).

Theorem

for Polynomials in

for Polynomials, J

x

A Theorem on Partial Fractions (302). lf

EXERCISE

Theorem

XXX

(304).

+ ...=0,

(303).

dx

+y

x and in

+ ...~nu dy

The Equation

CONTENTS

xii

CHAPTER

XIX. EXPONENTIAL AND LOGARITHMIC FUNCTIONS AND

SERIES.

The Exponential Continuity, Inequalities and Limits (306, 307). x Theorem, Series for a (307, 308). Meaning of an Irrational Index, Derivatives of a x log x and x n (309). Inequalities and Limits, the x way in which e and log x tend to oo Euler's Constant y, Series for 2 The Exponential Function E(z), Complex Index (310-312). log ,

,

(312, 313).

Series for sinx, cos x arid Exponential Values (313). in Summing Series (314).

Use of Exponential Theorem

EXERCISE

XXXI

(315).

Logarithmic Series and their Use in Summation of Series, Calculation of Logarithms (315-319). The Hyperbolic Functions (319-321).

EXERCISE

XXXII

XX. CONVERGENCE

(321).

(2).

Series of Positive Terms.

2

-

'

Cauchy's Condensation Test, Test Series

Rummer's, Raabc's and Gauss's Tests

(325).

Binomial and Hyper-geometric Series (328, 329). %

(326-328).

De Morgan and

Bertrand's Tests (330).

Terms Positive or Negative. Theorem, Abel's Inequality, and Abel's Tests (330, 331). Power Series, Interval and Radius of Convergence, Criterion for Identity of Power Series (332Binomial Series l-f?iz4-... when z is complex (334). Multi334). plication of Series, Merten's and Abel's Theorems (335-338). Series with

Dirichlct's

EXERCISE XXXIII

(338).

XXI. JBlNOMIAL AND MULTINOMIAL THEOREMS. Statement,

Vandermonde's Theorem

Binomial Theorem.

(340).

Euler's Proof, Second Proof, Particular Ins tancesj 34 1-345).

Num-

";

1 + x (345-348). erically Greatest Term, Approximate Values of EXERCISE XXXIV (349). Use of Binomial Theorem in Summing Series, ^Multinomia^Theorem

(351-355).

EXERCISE

XXXV

(355).

XXII. RATIONAL FRACTIONS

(2),

PR,ECURRING)SERIES

AND DIFFERENCE

EQUATIONS. Expansion of a Rational Fraction (357-359).

EXERCISE

XXXVI

(359).

Expansions of cos nd and sin nOjsin 9 in Powers of cos 6 (360). Recurring Series, Scale of Relation, Convergence, Generating FuncLinear Difference Equations with Constant tion, Sum (360-363). Coefficients (363-365).

EXERCISE

XXXVII

(365).

Difference Equations, General

EXERCISE

XXXVIII

(370).

and Particular Solutions

(367-370).

CONTENTS

xiii

CHAPTER

XXIII. THE OPERATORS 4> E, D. The Operators J, E, Terms of u l9 Au^ A 2 u l9

INTERPOLATION. A ru x u x+r

Series for ...

,

;

U1 + u2 + u3 +

...

in

and

Interpolation, Lagrange's

(373-379).

d \n (uv) ( j- ) Vcte/ /

Bessel's

The Operator

Formulae (379-382).

f) 9

Value of

(382, 383).

EXERCISE

XXXIX

(384).

XXIV. CONTINUED FRACTIONS

(1).

Definitions, ForniationjoConvergents, Infinite Continued Fractions

(388-391).

EXERCISE

Simple and ^eeuTrirSg; Continued Fractions (391-394).

XL

(394).

Simple Continued Fractions, Properties of the Convergents, an Irrational as a Simple Continued Fraction (396-401). Approximations, Miscellaneous Theorems (402-406). Symmetric Continued Fractions, Application to Theory of Numbers (406-409). Recurring Con tinned Fractions (409-411).

EXERCISE XLI

(411).

XXV. INDETERMINATE EQUATIONS

OF THE FIRST DEGREE.

axby axbycz...=k

Solutions of the Equation

EXERCISE XLII

Simple

c (414-417).

Two

Equations in

x, y, z

;

(417-419).

(419).

XXVI. THEORY OF NUMBERS

(2).

Congruence, Numbers less than and prime to n, Value of q)(n) E

^e resu

power of a prime p which

is

lt

follows.

n

contained in

is

\

For, of the numbers from 1 to n inclusive, there are I[n/p] which are 2 2 and so on hence the by p of these I[n/p ] are divisible by p

divisible

;

;

;

result follows.

9.

by

Theorems.

(1)

The product of any n consecutive

integers is divisible

\-n.

For (m + l)(w + 2) last expression is

occurs in

[m

n^

...

(m-f n)/\ n =

\

m + nj\ m

w,

and to show that the

it is sufficient to show that any prime p which Thus we occurs to at least as high a power in -f n.

an integer

m

j

have to show that /[ (m

+ n)/p] -f /[ (m + n)/p*] + 1[ (m -f n)/p 3] +

Now

/[(m + n)/p]>/[w/p]-f /[n/p], and the same is true by p p*, ... in succession hence the result in question. 2

ft

. . .

,

,

:

if

we

replace

NUMBERS IN ARITHMETICAL PROGRESSION

8 (2)

is

Ifn

a prime,

is divisible

C?

by n.

n(n - l)(n 2)

For by the preceding since n is a prime and r r is

Hence,

...

-r+

(n

1) is divisible

supposed to be less than n, r a divisor of (rc-l)(n-2) ... (n-r + 1) is

is

by

[/% aijid

prime to

;n.

|

|

and Thus

a prime,

t*

ft

t/

except the first

and

last,

text-books about

'

divisible

is

l)/[r

by

n.

in the expansion of (l+x)

all the coefficients

n ,

are divisible by n.

supposed to be acquainted with what is said in elementary for permutations and combinations and the binomial theorem

The reader

NOTE.

(n-r +

...

^(ft-l)

is

*

'

a positive integral index.' In what follows, F$ denotes the number of permutations, and C the number of combinations, of n things taken r at a time. Ex.

Find

1.

We have

the highest

power of 5 contained in

7[158/5]-31,

158. |

2

/[158/5 ]=/[31/5J-:6,

therefore the required power has an index

1

/|158/f>J=y|y or yc, or if (2) If a a>6 and 6^c, then a>c, (4) If a>b then -a< - 6. 2.

:

then 6 = a.

Hence we deduce the following rules for where it is assumed that zero is not used as a (5) If

b

+ x,

a-x = b-x

(6) If

a = b and x = y, then

(7) If

a>b

inequalities,

:

)

ax

bx

9

a/x

b/x.

then

and

ax^bx and a/x>b/x

(8) If

divisor

and

a= a+x

Also

equalities

a>b and x>y,

according as x

then a + x>b

both positive, then ax>by.

a-x>b-x. is

positive or negative.

+ y, and

if

a and y or b and x

ai!

REPRESENTATION OF NUMBERS BY POINTS Fundamental Laws of Arithmetic.

3.

Any two

13

rationals can

of addition, subtraction, multiplication and the result in each case /jvision, being a definite rational number, excepting lat zero cannot be used as a divisor. This is what is meant when it is

combined by the operations

j8

iid that the

system of rationals

The fundamental laws

;

f

(1)

(3)

4.

an

and multiplication are

+ 6 = 6 + a, ab = ba,

+ 6) + c = a + (6 + c), - ac + be, (a + b)c (a

(2)

(4)

(5)

The

closed for these operations.

a

a nd

fifth

is

of addition

= a(bc). (ab)c

and third constitute the Commutative Law, the second and

first

the Associative Law, the fourth the Distributive Law.

Theorem

integer

n

of Eudoxus.*

If a and b are any two positive rationals, nb>a. This simply amounts to saying that an greater than a/6.

exists such that

integer exists which

is

Representation of Numbers by Points on a Line. Take a

5.

OX

take a point 1 so that the segment 01 X'OX as axis in contains the unit of length. To find the point a which is to represent any Divide the segment 01 into n equal parts positive rational a, let a = m/n.

straight line

and

;

set off a length Oa, along

which represents - a

is

OX, equal

OX', at the

in

m

i

FIG.

The point

of these parts.

Ola

i

-a

X'

to

same distance from

as the point a.

I

i

X

1.

Points constructed thus represent the rational numbers in the following and one only. respects (i) For every number there is one point :

(ii)

The point point a

X

is generally taken to the right of to the right of the point 6.

Absolute Values.

6.

- a,

is

The points occur in the order in which the corresponding numbers stand on the rational scale.

according as a

|a-6|=|6-a|.

is

It

The

positive is

f

a

i is

or negative, and

x>a

is

denoted by

a>6, the

a

is

-fa or

.

Thus

a |

\

obvious that

positive, to say that

x |>a, then

if

absolute or numerical value of

+ 6, j

so that,

or

|

x
oo

Aggregate.

that x tends

is

below, or from the

9.

how

always greater than a, we say that x tends to a This is expressed by writing x -> a -f 0. right.

x tends to a and

from To say that x

to say

choose, no matter

may

x - a tends to we write x -> a. If

is

numerical value becomes and remains

than any positive number that we This

tends to zero

an

infinite set.

^^

APPROXIMATE VALUES 10.

System Everywhere Dense.

system of rationals

many

infinitely

For

if

is

An

important property of the that between any two rationals a and b there are

is

rationals.

a - l/u. The values of u and v may be in error by as much as 2 per cent, of the corresponding true values. If the true values of u and v are 20 and 13, show that the value of/ '

14.

as calculated from the observed values of its true value.

may

be in error as

much

as 9-9 per cent,

APPROXIMATE VALUES

22

The weight (w grammes) of water displaced by a solid is given by the w=w l -w 2 where w l9 w 2 are the weights (in grammes) of the solid (i) in

15.

formula

,

In determining the values of w i9 w 2 the error in each case per cent, of the true value. If the true values of w lf w 2 are 13 and 10, show that the value of w calculated from the observed values of w ti> a may be in error by as much as 7 per cent, of its true value. [The greatest and least possible values of w 1 are 13-13 and 12-87, those of w z are 10-1 and 9-9, hence those of w l -to* are 3-23 and 2-77.]

vacuo,

may

in water.

(ii)

be as

much

as

,

1

ly, then x-y*j(2x) of s/(# 2 - i/ 2 ) with an error in excess less than t/ 4 /(2a; 3 ). 2 [In Ex. 3 of Art. 17, put a = y /x*.]

a and b are nearly equal, show that - (a + 6)

18. If

_ value of Va6 with an error in excess

less

-

than

(

a

x

,

an approximate value

-

is -j~

_M4 r

is

...

4(d -}- uj

an approximate

[Use Ex.

3

17.]

19. Given that ^14 = 3-741..., 4/3-1-442..., for each of the following obtain a decimal approximation with an error in defect, finding also an upper limit to the error :

(i)

20. If

^14

0

For

(x

?

+ ax

//

an> token

)

(x

together.

p l9 p z p 39 ,

one, two, three,

+ a 2 ) (x + a 3 )

...

(x

4-

denote the

...

at

...

an )

i

sums of

the products of

a time, respectively, then

equal to the

products which can be formed by choosing a term out factors and multiplying these terms together,

sum of

of all

the

each of the

BINOMIAL THEOREM

34

n Choosing x out of each factor, we obtain the term x of the expansion. Choosing x out of any (n 1) of the factors and the a out of the remain-

we obtain

ing factor,

xn

~l

+ a 2 + d3 +

(a l

...

f an )

or

p^x*-

1 .

Choosing x out of any (n 2) of the factors and an a from each of the two remaining factors, we obtain xn

~

/y7t

2

"T~ /i'o*/

I i

~ n/y nffl~~T _1

jf

. I

I

"^

C*rI%

|r-j)|n-r

Thus

kr ^C^(aQ

which proves the theorem.

Multinomial Theorem for a Positive Integral Index. n The product -f k) Expansion of (a + 6 -f c +

16. (1)

. . .

(a is

the

sum

.

+ 6 + c + ... +k)(a + b + c + ...

-f

k)

...

to

w

factors

which can be formed by choosing a term out and multiplying these terms together. therefore the sum of a number of terms of the form

of all the products

of each of the

n

factors

The expansion is ... k, where each index may have any

aab^cy

of the values 0,

1, 2, ...

,

w,

subject to the condition ...-fic

= tt

(A)

MULTINOMIAL THEOREM

36

Choose any positive integral or zero values for

To obtain the

satisfy this condition.

a,

coefficient of the

...

y,

jS,

term a

K which

,

a

b^cv

...

k" y

K factors, respecdivide the n factors into groups containing a, )S, y, of of factor the first a out each b out of each factor of Take group, tively. . . .

,

the second group, and so on.

The number

of

ways

in

which

this

can be done

is

In

and

which

of times

7

a a b^cy

this is the coefficient of

term

this particular

(B)

,

a ...

k" in the expansion

.aWcv where a,

from

y,

jS,

...

K are

,

numbers

the

to

have ...

0, 1, 2,

,

all possible sets

n,

subject to the

Jt,

(C)

of values which can be chosen condition

Theorem

called the Multinomial

is

...

+K = n.

a + j8-f y-f... This result

number

the

(i.e.

Hence

occurs).

for a positive integral

index.

Thus

(2)

in the expansion of (a

If there are

m numbers a,

+ b 4- c -f

sion of (a

. . .

-f

- b - c -f d) 6 the

b, c, ... k,

coefficient of a*b 2 c is

the greatest coefficient in the

expan-

n k) is

and r the remainder when n is divided by m. n I a ^ y The coefficient \K has its greatest value when a j8 has its least value. Denote this value by v, then v is the least value * if a and j3 alone vary and y, K are constant. a j8 where q

is the quotient

. . .

. . .

l/c

|

|

|

\

|

|

|

Thus

a

C

j8

must have

value subject to the condition a

its least

|

|

where

is

4-

j8= C,

a constant.

Let wa =|a|/J = |a|C-a, then ing as

of

. . .

. . .

|

|

a^C~a + l,

when a=j8=4n,

*

25. If

prove that

(i)

a r = 2 r n {C* n

+ CjCj n C"r

the last term in the bracket being or odd.

"2

or (^

+ C^Cf

at

(iii)

=4n

= \V-

[We have

17.

(1

l l ii(l+3C- + 5C%- + ...

--2

n

n(n -]){!. 3

4-2x4- 2x

Highest

2

4-3

2

5C?"

.

accorc^ n g as r

4-5

7f7>

.

is

cven

n terms).

to

l

~2 4- ...

2

to (n - 1) terms}.

)^

Common

in a single variable x.

+ ...},

+ l)^7(H-i)

In particular, show that 2 2-n (ii)a 2

~4

Factor (H.C.F.). These will be denoted by

We consider polynomials capital letters.

where A, B, Q, R are polynomials, the common factors B are the same as those of B and R. For since A BQ 4- R, every common factor of B and R is a factor of A and, since R~A BQ, every common factor of A and B is a factor (1)

of A

of

// A = BQ + R,

and

7?.

(2)

(Q 2

,

If

R2

),

B

is

not of higher degree than A, polynomials or constants (Q }9 Rj), ... can be found such that

($ 3 J? 3 ) ,

where the degree of any one of the

set

B,

R R2 l

,

,

is less

...

than that of the

preceding.

from the

It follows (a) if

factor, this is a (h) if

theorem that

last

R2

one of the pairs (A, B), (B, RJ, (R v

common

one pair have no

factor of every pair

common

),

(R2

,

R3

)

have a

common

;

factor, the same

true for every pair.

is

Moreover, the process must terminate and that in one of two ways the last remainder, say R n must vanish identically, or be independent of x. :

,

Rn

If

Q,

then

R n _ 2 = Rnnl Q n

,

so that

Rn ^

is

a .common factor of

(7?n _ 2 ^n-i)> an d therefore also of (A, B). Moreover, every common factor of (A, B) is a factor of J? n _j, which is therefore the H.C.F. of A, B.

If

Rn is independent of x, then

the same (3)

is

(J? n _ 2 ,

Rn -i)

have no

common

factor,

and

true of (A, B).

The H.C.F. Process.

by successive

Equations

(1)

of the last section are

divisions, as represented below.

found

H.C.F.

In practice, the reckoning

B) A

is

PROCESS

41

generally arranged as on the right.

(Q,

A

B

BQi

(Q 2

Qi

BQ, etc.

etc.

The process may often be shortened by noticing that

:

multiply or divide any of the set A, J5, R l9 R 2 ... by any constant or by any polynomial which is not a factor of the preceding member of the set. (i)

We may

(ii)

If

we

,

arrive at a remainder

Rn

which can be completely factorised,

the process need not be continued. For factors is a factor of R n __ v we can write

(iii)

we find which, if any, of these down all the common factors of

if

We may use the following theorem X^lA + mB and Y ^I'A + m'B, where :

m, I', m' are constants Z, the H.C.F. of X and Y and such that then Im'-l'm^Q, different from is the same as that of A and B. For every common factor of A and B is a common factor of X and Y. //

zero

Moreover,

A(lm'-Vm) = m'X-mY therefore every Ex.

is

1.

Find

common factor

the H.C.F. of

Paying attention to the as follows

of

and

(i)

r

X and Y is a common factor of A and B.

A = 3#3 +# + 4

remark

B(lm -l'm) = lY ~-VX,

and

B = 2x*

and using detached

coefficients, the

reckoning

:

3+0+ 1+

4

(a)

Hence the H.C.F. =x + l. But it is unnecessary to go beyond the step marked (a), which shows that E t = 3x 2 + 2x - 1 = (3z - 1 (x + 1 ). Now x + 1 is, and 3x - 1 is not, a factor of B. Therefore x + 1 is the H.C.F. of B, R 19 and consequently that of A and B. )

PRIME AND COMPOSITE FUNCTIONS

42 Ex.

Find

2.

A and B where, ^8*5 + 5* + 12. A = 12x5 + 5x* + S and 2 A - 3B = 10*3 - 15z 2 - 20 - 5 (2* 3 - 3* 2 - 4), 3 A -2B = 20x 5 + 15*3 - Wx* = 5x*(4:X* -f 3x - 2).

the H.C.F. of

2

We have

is

3 2 3 Putting C = 2x -So; -4 and Z> = 4# the same as that of C and D.

-

2, it

A and B

follows that the H.C.F. of

C - 2D - 6z 2 + 3z + 6 = 3 (2z 2 + # + 2), 2(7 - D = 6x* + 3* 2 + 6z = 3a? (2x 2 + x -f 2).

Further,

of

+ 3x

Hence the H.C.F. of C -2D and 2C C and D and also that of A and 5.

-D

is

2x 2 + x + 2

:

this is therefore the H.C.F.

t

Prime and Composite Functions.

18.

factors except itself (and constants)

it is

a polynomial has no said to be prime otherwise it is If

:

said to be composite.

Thus x 2 + 3x -f 2 is a composite function whose prime factors are x 4- 1 and x + 2. Polynomials which have no common factor (except constants) are said to be prime to each other. Such expressions have no H.C.F. Thus 2(x + 1) and 4(cc 2 -f 1) are prime to each other.

We

can prove theorems for polynomials analogous to those of Ch. and 5, relating to whole numbers.

I,

Arts. 3, 4

Remembering the

between arithmetical and algebraical

distinction

primeness, the reader should have no difficulty in making the necessary verbal alterations. Thus, the theorem corresponding to that of 4 (i) is as follows

:

M

M

are polynomials and If A, JB, is a factor of B.

For

if

H.C.F. of

common

M

is

prime

is

a factor of AB and prime

prime to A, these two have no

common

to

A, then

Hence the

factor.

MB and AB B. But M a factor of AB, and therefore a MB and AB. Hence M equal to, or a factor of, B. is

is

is

factor of

A

is

is

B

and are polynomials in x, then polynomials If each other, can be found such that

Theorem. to

M

AX + BY = l according as

A

is or is not

prime

to

or

AX + BY = G,

B G 9

X, Y,

being the H.CJF. of

A

and

B

in the

latter case.

For

if

Q 19 Q2

,

...

are the quotients

process of searching for the H.C.F. of

and B l9 J?2 A and B,

,

...

the remainders in the

etc.

IMPORTANT USE OF we obtain

Therefore

where

in succession the following equations

we can

Continuing thus,

the last remainder

Hence we can

express any remainder in the form

find

Y

X,

AX + BY,

is

either G, the H.C.F.

In either case

X+

.

-f,

common

X

Y = 1 and G

factor of

or

it is

a constant

c.

or

prime to Y, for the

is

;

such that

AX+BY^G .

:

X and Y are polynomials.

Now

-^

43

H.C.F.

is

a factor of

X and Y would

A

and

first

of

equation

J3.;

may

be written

thus in either case any

be a factor of a constant.

In the second case, the polynomials in question are X/c and Y/c. NOTE.

This theorem

is

fundamental

in the

Theory of Partial Fractions, which

are dealt with in Ch. VII.

EXERCISE V H.C.F.

Find the

AND

H.C.F. of the functions in

ITS

Exx.

USE

1-6.

2. 3.

2z 4 - 13z 2 + x + 5

15,

3z 4 - 2z 3 - 11 x 2 + I2x + 9.

4.

z -f5* -2, 2# -5o; 3 -fl.

6.

12z 3 + 2* 2

In Exx.

5

2

5.

2x*-5x 2 + 3, 3z 5 - 5x3 -f 2.

-2LK-4, 6*3 -t-7* 2 -14a;-8, 2Lr 4 -28a; 3 -f 9* 2 -

7-9, find

16.

polynomials X and Y for which the statements are identically

true. 7.

(*-l)

2

Z-(* + l) 2 F=:l.

8.

9.

10.

Use the

and hence

find

H.C.F. process to obtain

A, B, C, D, such that

Ax + B

Cx + D

[Multiply the first identity by A, the second by /*, and add that (7A + 10 Li)/(llA + 28^)=2/(-3); and thus obtain

:

find A

and

p,

such

/

B.C.A.

CHAPTER

IV

SYMMETRIC AND ALTERNATING FUNCTIONS, SUBSTITUTIONS

A

Symmetric Functions.

1.

any two

interchange of

symmetric with regard

function which

of the variables

which

it

is

unaltered by the is said to be

contains

to these variables.

2 2 2 2 2 y + y z 4- z x) (x z -f y x + z y) are symmetric with regard to x, y, z. (In the second expression, the interchange of any two letters transforms one factor into the other.)

Thus yz + zx + xy and

2

(x

The interchange of any two letters, x, y, is called the transposition (xy). Terms of an expression which are such that one can be changed into the other by one or more transpositions are said to be of the same type. Thus all the terms of x 2y -f x 2 z -f y 2z + y2 x + z2 x -h z2 y are of the same type, and the expression is symmetric with regard to x, y, z. symmetric function which is the sum of a number of terms of the

A

same type

is

often written in an abbreviated form thus

of the terms and place the

x

-f

Again, (x It

y+z

is

letter

Z (sigma)

Hx and

represented by

For instance,

it.

before

yz

Choose any one

:

+ zx + xy by Zxy.

+ y -f z) 2 = x 2 + f + z 2 + 2yz + 2zx + Zxy = Zx2 + 2Zyz.

obvious that

is

If a term of some particular type occurs in a symmetric function, then

(i)

terms of this type occur.

all the

The sum,

(ii)

difference, product

and

quotient of two symmetric functions

are also symmetric functions.

Considerations of

symmetry greatly facilitate many algebraical

processes,

as in the following examples. Ex.

1

.

Expand

(y

-f

z

- x)

(z

+x

-

y) (x

-f

y

- z).

symmetric, homogeneous, and of the third degree in - x) + x - (x + ~ = a Zx3 + b therefore assume that (y + z y) y z) (z

This expression

may

is

.

where

a, 6, c, are independent of In this assumed identity,

.

x,

y

>

z.

We

2x zy + cxyz,

x, y, z.

then -1 a; 0, z (i) put x~ 1, y = 2a + 26, .'. (ii)puts = l y = l,z = 0; then a? = l, z 1 then l=3a + 66+c, ; 2/-1, (iii) put ;

f

Hence the required product

is

- a:3 - y3 -

z

3

-f

y-z

6^1; /. -f

-2.

c

yz-

-f

z~x

-f

zx 1

-f

x 2y

4-

xy

2

-

2xyz.

ALTERNATING FUNCTIONS Ex.

Expand

2.

(a

+ b+c+d)(ab + ac+ad + bc + bd + cd).

45 Test the result by putting

a~b=c~d = \. The product is the sum of all the products which can be obtained by multiplying any term of the first expression by any term of the second. Hence the terms in the product are of one of the types a?b abc. The coefficient of a z b in the product is t

1

and ab, and in no other way. The coefficient of abc is 3 for this term

;

for this

term

is

obtained as the product

of a

;

is

obtained in each of the three ways a(bc),

b(ac) 9 c(ab).

a 2 b + 32abc. Hence, the required product is The number of terms of the type a-b is

Test.

type abc

4

is

hence,

;

a

if

Ea Sab ^ 4 .

so that the test

Ex. 3

c

G

= 24

12,

and the number of terms of the

,

and

Za-b

\-

%abc = 12 + 3

.

4 - 24,

is satisfied.

Factor ise

.

.

~d = 1

b

(x

-f

y

-f

- x5 - 5 - 2 5 y

5

z)

.

Denote the given expression by E. Since E-0 when x - - y, it follows that x + y The remaining factor is a is a factor of E; similarly, y + z and z + x are factors. homogeneous symmetric function of #, y, z, of the second degree. We therefore assume that

where a and

b arc

independent of put x~l,

?/--!,

put x^\, y

2.

and proceed thus then 2a + b = 15. z-0

x,

//,

z,

:

;

= l, z^l

Alternating Functions.

;

/.

,

a ^-5

6=5

a + b^-lQ.i

then

If a function

of x, y,

z, ... is

trans-

formed into - E by the interchange of any two of the set x, y, z, ... then E is called an alternating function of x, y, z, ... n for the interchange of z n (x Such a function is x n ,

.

(y-z)+ y (z-x)+

any two

letters,

say x and

y,

n

transforms n

y)

it

n

y (x~z)+x (z-y)+z (y-x)

Observe that

the product

and

;

into

= -E.

the quotient of two alternating functions are

symmetric functions. 2

Thus

{x

3 ~ (y-z) + y (z-x)+ z* (x y)}/ (y -z)(z- x} (x y)

with regard to

is

symmetric

x, y, z.

Ex.

1. Factorisex*(y-z)+y*(z-x)+ z*(x-y). Denote the expression by E. Since E ~0 when x = y, Similarly y -z and z -x are factors, thus

it

follows that x

-y

is

a

factor.

E = (y-z)(z-x)(x-y).F, where

F

where k

is

symmetric, homogeneous and of the

is

independent of - 1, thus

find that k

x, y, z.

E=

first

Equating the

degree in x,

y, z.

coefficients of x*y

-(y-z)(z-x)(x-y)(x+y + z).

Hence

on each

side,

we

IMPORTANT IDENTITIES

46 3.

An

expression is said to be cyclic with k, arranged in this order, when it is un-

Cyclic Expressions.

regard to the letters a, 6, c, of, ... h, altered by changing a into b, b into

c into d,

c,

. . .

,

h into

k,

and k into

a.

This interchange of the letters is called the cyclic substitution (abc ... k). Thus a 2 b + 62 c -h c2d -f d2 a is cyclic with regard to a, b, c, c? (in this order), for the cyclic substitution (abed) changes the first term into the second, the second into the third, It is obvious that

. . .

,

and the

last into the first.

If a term of some particular type occurs in a cyclic expression, then the term which can be derived from this by the cyclic interchange, must also occur ; (i)

and

the coefficients of these terms

(ii)

The sum,

must be

difference, product,

and

equal.

quotient, of two cyclic expressions are

also cyclic.

In writing a cyclic expression, Thus X2 (y-z) + y2 (z-x)+z 2 (x-y)

unnecessary to write the whole. 2 is often denoted by Zx (y-z), where the meaning of 27 must not be confused with that in Art. 1. 2 2 2 Again, we sometimes denote x (y -z) + y (z-x)+z (x-y) by is

it

x2 (y-z)

+

.

..

+

...,

being understood that the second term is to be obtained from the and the third from the second by cyclic interchange.

it

The student should be

familiar with the following important identities.

2.

a(&-

3.

a 2 (b-c)+b 2 (c-a) + c2 (a-b)= -(b-c)(c-a)(a-b).

4.

bc(b-c)-\- ca(c-a)

5. 6.

a(b

2

2

-c ) + b(c

3

a (i-c)

2

first

+ ab(a-b) = -(b-c)(c-a)(a-b). -a + c(a 2 -b 2 = (b-c)(c-a)(a-b). 2

)

)

+ 6 3 (c-a) + c 3 (a-6)= -(b-c)(c~a)(a-b)(a + b + c).

7.

8. 9.

10. 11.

a+&+

It will be (i)

(ii)

Any

c& + c-ac + a-ba + b-c)=

proved later

that

symmetric function of a,

Any symmetric

27a, Za/J,

(pp. 95, 96)

Za/?y and

aj

-a* -b* -

function of a,

j8,

j8,

y can be expressed y, 8

in terms of

can be expressed in terms of

SUBSTITUTIONS mode

This

functions,

of expression is

and

47

extremely useful in factorising symmetric

in proving identities.

-6 2 )(1 -c 2 )+ 6(1 -c 2 )(l -a 2 )+ c(l -a 2 )(l -6 2 )-4a6c. Denoting the given expression by E we have Ex.

1.

Factorise

a(l

t

= Sa - Zab* + abcZab - 4abc. Za6 2 = Za Zab - 3abc E Za - Za Zab - abc + abcZab

Now

.

4. Substitutions.

ment

;

.

.*.

= (1

-be -ca-a&)(a + b+c-abc).

(1)

We

of a set of elements

may

consider processes

by which one arrange-

be transformed into another.

Taking the permutations cdba, bdac of a, 6, c, d, the first is changed into the second by replacing a by c, 6 by a, c by 6 and leaving d unaltered. This process

is

represented by the operator

/abc\

fabcd\ , \cabdJ f

or

)

,

)

.

(abc\Icaoa = bdac. , \cabj

and we write

,

(

\cab/

Such a process and

As previously

also the operator

which

stated, the interchange of

effects it is called a substitution.

two elements

a, 6 is called the

transposition (ab).

Also a substitution such as

fabcd\ _

(

in

),

which each

the one immediately following it and the last by the substitution or cycle, and is denoted by (abed). If is

letter is replaced first, is

called a cyclic

two operators are connected by the sign = the meaning ,

equivalent to the other, thus (abcd)

is

Let

that one

= (bcda).

Two

or more substitutions may be applied successively. as indicated follows, the order of operations being from right to left. (2)

by

This

is

S = (a&), T = (6c), then ^ Sacbd = bcad, and om = fabcd\ ST

mi

Thus

(bcad)>

This process substitution

is

is

TS = (abcd ,

called multiplication of substitutions,

and the

resulting

called the product.

Multiplication of this kind substitutions have

no common

is

not necessarily commutative, but if the

letter, it is

commutative.

The operation indicated by (aft) (aft), in which (ab) is performed twice, produces no change in the order of the letters, and is called an identical substitution.

SUBSTITUTIONS

48

substitution is cyclic, or is the product of two or

Any

(3)

cyclic sub-

which have no common element.

stitutions

As an

more

instance, consider the substitution

fabcdefghk\

s

\chfbgaedk/

Here a

c, c to /, / to a, thus completing the cycle (ac/). to h to d, d to 6, making the cycle (bhd). Next, e is h, changed to and to The element k is unchanged, e> giving the cycle (eg). g changed g and we write

Also b

is

changed to

is

S = (acf)(bhd)(eg).

or

S~(acf)(bhd)(eg)(k)

This expression for S is unique, and the order of the factors is indifMoreover, the method applies universally, for in effecting any substitution we must arrive at a stage where some letter is replaced by the ferent.

thus completing a cycle. The same argument applies to the set of not contained in this cycle.

first,

letters (4)

A

cyclic substitution of

product

ofn-1

transpositions.

(abo)

have equalities such as

also

(ae) (ad) (ac) (ab) (5)

is the

= (ab) (be), = (abc) (cd) = (ab) (be) (cd), (abed) = (abed) (de) = (ab) (be) (cd) (de), and so on. (abcde)

For

We

n elements

A

= (abcde)

(ab) (ac) (ad) (ae)

,

n

substitution which deranges

cycles is equivalent to

n-r

letters

= (edcba)

and which

is the

.

product of r

transpositions.

This follows at once from

(3)

and

(4)

.

Thus

if

S=(

^

a

ef^ h

]

\chfbgaed/

S = (ac/) (bhd) (eg) = (ac) (c/) (bh) (hd) (eg)

then

we

introduce the product (a6)(a&), transpositions is increased by 2. If

Thus,

a given substitution

if

number j

is

not unique.

j=n This

(6)

and

unaltered and the

number

of

equivalent to j transpositions, the prove that

is

shall

- r + 2s where

is

s is

a positive integer or

a very important theorem, and to prove

is

notion of

We

S

.

zero.

it

we introduce the

*

inversions.'

Taking the elements a, 6, c, d, the normal arrangement.

e,

choose some arrangement, as abcde,

call it

Consider the arrangement bdeac. Here 6 precedes a, but follows it in the normal arrangement. On this account we say that the pair ba con-

INVERSIONS stitutes

an

inversion.

Thus bdeac contains ba,

da,

dc,

49

five inversions,

ea,

namely,

ec.

Theorem 1. If i is the number of inversions which are introduced or removed by a substitution which is equivalent to j transpositions, then i and j are both even or both odd.

For consider the

a single transposition (fg). If f, g are consecutive elements, the transposition (fg) does not alter the position of/ or of g relative to the other elements. It therefore introduces effect of

or removes a single inversion due to the interchange of /, g. Iff, g are separated by n elements p, q, r, ... #,- then/ can be

the place occupied by g by n + 1 interchanges of consecutive elements, and then g can be moved to the place originally occupied by / by

n such

interchanges.

moved

to

fpq - x