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

Answer:

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

Step-by-step explanation:

We already know that the partial fraction decomposition of the rational fraction $\frac{x-9}{x^2-3x-18}$ has a particular form, that is

$\frac{x-9}{x^2-3x-18} = \frac{A}{x-6}+\frac{B}{x+3}$.

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

First: We calculate the sum $\frac{A}{x-6}+\frac{B}{x+3}$.

So,

$\frac{A}{x-6}+\frac{B}{x+3} = \frac{Ax+3A+Bx-6B}{(x-6)(x+3)} = \frac{(A+B)x +(3A-6B)}{x^2-3x-18}$.

Notice that

$\frac{x-9}{x^2-3x-18} =\frac{(A+B)x +(3A-6B)}{x^2-3x-18}$,

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 $B=4/3$

Answer 2

Answer:

Sorry I read in class 9 standard