Understanding Engineering Mathematics

where theA,B,Care ‘unknown’. In fact, the only way the left- and right-hand sides can
beidenticalis if the coefficients of corresponding powers are the same, i.e.

A= 1 ,B=− 3 ,C= 2

Another way of looking at this is to say that since the identity must hold true for all values
ofx, we can determine theA,B,Cby substituting ‘useful’ values ofx. For examplex= 0
givesC=2 immediately. Choosing any two other values forxwill give two equations
forAandB. In this case we may notice that the roots of the quadratic are 1 and 2 and so
taking these values ofxgive

x= 1 ,A+B+ 2 = 0

x= 2 , 4 A+ 2 B+ 2 = 0

Solving these simultaneous equations gives

A= 1 ,B=− 3 , as before.

Solution to review question 2.1.5

To determineAandBin

A(x− 3 )+B(x+ 2 )≡ 4

use the fact that this must be true forallvalues ofx. In particular it must

hold ifx=3, giving

A( 0 )+B( 5 )= 4

or

B=^45

Similarly, puttingx=−2gives

A(− 5 )= 4

or

A=−^45

An alternative (longer) approach is to rewrite the identity as

(A+B)x+(− 3 A+ 2 B)= 4

and equate coefficients on each side to give

A+B= 0

− 3 A+ 2 B= 4

Now solve these to give the previous results.