Question

The partial fraction decomposition of LaTeX: \frac{x-9}{x^2-3x-18} x − 9 x 2 − 3 x − 18 is LaTeX: \frac{A}{x-6}+\frac{B}{x+3} A x − 6 + B x + 3 . Find the numbers LaTeX: A\: A and LaTeX: B B . Then, find the sum LaTeX: A+B A + B , which is a whole number. Enter that whole number as your answer.

Answer 1

Not entirely sure what the question is supposed to say, so here's my best guess.

First, find the partial fraction decomposition of

`(x-9)/(x^2-3x-18)`

This is equal to

`(x-9)/((x-6)(x+3))=\frac a{x-6}+\frac b{x+3}`

Multiply both sides by
`(x-6)(x+3)`, so that

`x-9=a(x+3)+b(x-6)`

Notice that if
`x=6`, the term involving
`b` vanishes, so that

`6-9=a(6+3)\implies a=-\frac13`

Then if
`x=-3`, the term with
`a` vanishes and we get

`-3-9=b(-3-6)\implies b=\frac43`

So we have

`(x-9)/(x^2-3x-18)=-\frac1{3(x-6)}+\frac4{3(x+3)}`

I think the final answer is supposed to be
`a+b`, so you end up with 1.