constants equation?

Question

find the constants A and B that make the given equation true.

x -16
————
x - 2x - 8

=

A
_____
x + 2

+

B
____
x - 4

I know it's written out really weird but any help at all would be greatly appreciated.

Answer 1

Multiply the first by (x-4)/(x-4) and the second by (x+2)/(x+2) i.e. the factors are both 1 where the function is defined. This allows addition of the numerators (top bits).

It is also possible to ignore the denominators as they are the same everywhere. This gives

x-16 = A(x-4) + B(x+2)

x-16 = x(A+B) -4A+2B

comparing similar x terms gives


A+B=1
-4A+2B=-16

Multiply the first simultaneous equation by 4 then add to get

6B = -12 or B=-2
leaving A=3

Answer 2

by using the theory of partial fractions, you can see that if you by clearing the denominator, you end up with

and equation that looks like.

x-16= a(x-4) + B(x+2)..

here you can let x=4. to get A=-2 and
let x= -2 to get B= 2/3

Answer 3

A = (-2 - 16)/(-2 - 4) = 3
B = (4 - 16)/(4 + 2) = - 2