175 49 27MB
English Pages 601 [605] Year 1952
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
LONDON
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 Skewsymmetric Determinants, Pfaffian (138
Suffix,
143),
ExERtad XVI
(143)
X. SYSTEMS OF EQUATIONS. Systems (149, 150). Linear Equations in Line at Infinity (150152). Linear Equations in Three Unknowns, Equation to a Plane, Plane at Infinity (153157). Definitions, Equivalent
Two Unknowns, EXEKCISE XVII
(158).
Systems of Equations of any Degree, Methods of Solution for Special Types (160164).
EXERCISE XVIII
XL
(164).
RECIPROCAL AND BINOMIAL EQUATIONS. The Equation Reduction of Reciprocal Equations (168170). xn  1=0, Special Roots (170, 171). The Equation xn A =0 (172). The Equation a 17  1 ==0, Regular 17sided Polygon (173176). 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> af 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 (187189). Ferrari's Solution and Deductions (189191). Descartes' Solution (191). Conditions for Four Real Roots (192ty). 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 (204209). 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 (ab)$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
(228232).
E*
ponential Inequalities and Limits, /
l\m
/
i\n
and
1) >(!+) m/ n/ \ /
1 \
n
lim (1fnj n_ >00 V
EXERCISE
XXIV
l\m /
/
=e,
nj
(1)
if
m>n,
(232,233).
(233).
General Principle of Convergence (235237). Limits of Inde termination (237240).
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 + ...Ha 1 6 n )/n=o6,
(240243).
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
(248254).
(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 (256261). 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 (261263).
XXVII
EXERCISE
(264).
^
XVII^CoNTiNuous VARIABLE.
Theorems Meaning of Continuous Variation, Limit, Tending to on Limits and Polynomials (266268) 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 (273277). 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 (280282).
EXERCISE XXVIII
(282).
Inverse Functions, Bounds of a Function, Rolle's Theorem, MeanValue Theorem (284288). 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
.
(299302).
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 (310312). 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 (315319). The Hyperbolic Functions (319321).
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 Hypergeometric Series (328, 329). %
(326328).
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 lf?iz4... when z is complex (334). Multi334). plication of Series, Merten's and Abel's Theorems (335338). 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 1345).
Num
";
1 + x (345348). erically Greatest Term, Approximate Values of EXERCISE XXXIV (349). Use of Binomial Theorem in Summing Series, ^Multinomia^Theorem
(351355).
EXERCISE
XXXV
(355).
XXII. RATIONAL FRACTIONS
(2),
PR,ECURRING)SERIES
AND DIFFERENCE
EQUATIONS. Expansion of a Rational Fraction (357359).
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 (360363). Coefficients (363365).
EXERCISE
XXXVII
(365).
Difference Equations, General
EXERCISE
XXXVIII
(370).
and Particular Solutions
(367370).
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
(373379).
d \n (uv) ( j ) Vcte/ /
Bessel's
The Operator
Formulae (379382).
f) 9
Value of
(382, 383).
EXERCISE
XXXIX
(384).
XXIV. CONTINUED FRACTIONS
(1).
Definitions, ForniationjoConvergents, Infinite Continued Fractions
(388391).
EXERCISE
Simple and ^eeuTrirSg; Continued Fractions (391394).
XL
(394).
Simple Continued Fractions, Properties of the Convergents, an Irrational as a Simple Continued Fraction (396401). Approximations, Miscellaneous Theorems (402406). Symmetric Continued Fractions, Application to Theory of Numbers (406409). Recurring Con tinned Fractions (409411).
EXERCISE XLI
(411).
XXV. INDETERMINATE EQUATIONS
OF THE FIRST DEGREE.
axby axbycz...=k
Solutions of the Equation
EXERCISE XLII
Simple
c (414417).
Two
Equations in
x, y, z
;
(417419).
(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^
...
(mf 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 (rcl)(n2) ... (nr + 1) is
is
by
[/% aijid
prime to
;n.


and Thus
a prime,
t*
ft
t/
except the first
and
last,
textbooks 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.
(nr +
...
^(ftl)
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=yy 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,
ax = bx
(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.
ax>bx. 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
a6=6a.
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 99 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 1313 and 1287, those of w z are 101 and 99, hence those of w l to* are 323 and 277.]
vacuo,
may
in water.
(ii)
be as
much
as
,
1
ly, then xy*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 = 3741..., 4/31442..., 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%
rj)nr
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 + j8f yf... 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 = aCa, 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
42x4 2x
Highest
2
43
2
5C?"
.
accorc^ n g as r
45
7f7>
.
is
cven
n terms).
to
l
~2 4 ...
2
to (n  1) terms}.
)^
Common
in a single variable x.
+ ...},
+ l)^7(Hi)
In particular, show that 2 2n (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 ^ni)> 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'XmY 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
16.
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.
2LK4, 6*3 t7* 2 14a;8, 2Lr 4 28a; 3 f 9* 2 
79, 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
yz
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 ab 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
Zab
\
%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 E0 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. z0
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 ,
.
(yz)+ y (zx)+
any two
letters,
say x and
y,
n
transforms n
y)
it
n
y (x~z)+x (zy)+z (yx)
Observe that
the product
and
;
into
= E.
the quotient of two alternating functions are
symmetric functions. 2
Thus
{x
3 ~ (yz) + y (zx)+ z* (x y)}/ (y z)(z x} (x y)
with regard to
is
symmetric
x, y, z.
Ex.
1. Factorisex*(yz)+y*(zx)+ z*(xy). 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 = (yz)(zx)(xy).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
(yz)(zx)(xy)(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 (yz) + y2 (zx)+z 2 (xy)
unnecessary to write the whole. 2 is often denoted by Zx (yz), 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 (zx)+z (xy) by is
it
x2 (yz)
+
.
..
+
...,
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 (bc)+b 2 (ca) + c2 (ab)= (bc)(ca)(ab).
4.
bc(bc)\ ca(ca)
5. 6.
a(b
2
2
c ) + b(c
3
a (ic)
2
first
+ ab(ab) = (bc)(ca)(ab). a + c(a 2 b 2 = (bc)(ca)(ab). 2
)
)
+ 6 3 (ca) + c 3 (a6)= (bc)(c~a)(ab)(a + b + c).
7.
8. 9.
10. 11.
a+&+
It will be (i)
(ii)
Any
c& + cac + aba + bc)=
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 caa&)(a + b+cabc).
(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
ofn1
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
nr
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