Question

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

Answer 1

Sorry I read in class 9 standard

Answer 2

The numbers are
`A=-1/3` and
`B=4/3`, and the sum of
`A+B` is 1.

Explanation:

We already know that the partial fraction decomposition of the rational fraction
`(x-9)/(x^2-3x-18)` has a particular form, that is

`(x-9)/(x^2-3x-18) = (A)/(x-6)+(B)/(x+3)`.

So, the method to find the coefficients
`A` and
`B` is:

First: We calculate the sum
`(A)/(x-6)+(B)/(x+3)`.

So,

`(A)/(x-6)+(B)/(x+3) = (Ax+3A+Bx-6B)/((x-6)(x+3)) = ((A+B)x +(3A-6B))/(x^2-3x-18)`.

Notice that

`(x-9)/(x^2-3x-18) =((A+B)x +(3A-6B))/(x^2-3x-18)`,

which means that necessarily

`x-9 =(A+B)x +(3A-6B)`.

Second: We equalize the coefficients of the same powers of
`x`.

The last equality we have obtained means that

`x=(A+B)x` and
`-9 = 3A-6B`.

From the above statement we deduce that
`A+B=1`.

Third: We obtain a linear system of equations, with the unknowns
`A` and
`B`.

`\begin{cases} A+ B & =1 \\ 3A-6B &= -9\end{cases}`

`3A-6B = -9`

The solutions to these system of equations are
`A=-1/3` and
`B=4/3`