Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.5 BINOMIAL EXPANSION

x=−1 gives respectively

52

24

=

C

6

−

D 1

2

+

2

4

,

36

7

=

B+C

7

−D 1 +2,

86

63

=

C−B

7

−

D 1

3

+

2

9

.

These equations reduce to

4 C− 12 D 1 =40,

B+C− 7 D 1 =22,

− 9 B+9C− 21 D 1 =72,

with solutionB=0,C=1,D 1 =−3.
Thus, finally, we may rewrite the original expressionF(x) in partial fractions as

F(x)=x+2+

1

x^2 +6

−

3

x− 2

+

2

(x−2)^2

.

1.5 Binomial expansion

Earlier in this chapter we were led to consider functions containing powers of

the sum or difference of two terms, e.g. (x−α)m. Later in this book we will find

numerous occasions on which we wish to write such a product of repeated factors

as a polynomial inxor, more generally, as a sum of terms each of which contains

powers ofxandαseparately, as opposed to a power of their sum or difference.

To make the discussion general and the result applicable to a wide variety of

situations, we will consider the general expansion off(x)=(x+y)n,wherexand

ymay stand for constants, variables or functions and, for the time being,nis a

positive integer. It may not be obvious what form the general expansion takes

but some idea can be obtained by carrying out the multiplication explicitly for

small values ofn. Thus we obtain successively

(x+y)^1 =x+y,

(x+y)^2 =(x+y)(x+y)=x^2 +2xy+y^2 ,

(x+y)^3 =(x+y)(x^2 +2xy+y^2 )=x^3 +3x^2 y+3xy^2 +y^3 ,

(x+y)^4 =(x+y)(x^3 +3x^2 y+3xy^2 +y^3 )=x^4 +4x^3 y+6x^2 y^2 +4xy^3 +y^4.

This does notestablisha general formula, but the regularity of the terms in

the expansions and the suggestion of a pattern in the coefficients indicate that a

general formula for powernwill haven+ 1 terms, that the powers ofxandyin

every term will add up tonand that the coefficients of the first and last terms

will be unity whilst those of the second and penultimate terms will ben.