Answer
$\frac{-1}{x+2} + \frac{1}{x-1}$
Work Step by Step
We must first find the Partial Fraction Decomposition:
$\frac{A}{x+2} + \frac{B}{x-1} = \frac{3}{(x+2)(x-1)}$
We must then solve for the constants:
$A(x-1) + B(x+2) = 3$
$Ax - A + Bx + 2B = 3$
$Ax + Bx = 0$
$-A + 2B = 3$
This can be represented with the following matrix:
$\begin{bmatrix}
1 & 1 & |0\\
-1 & 2 & |3\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & |0\\
0 & 3 & |3\\
\end{bmatrix}$
$3B = 3$
$B = 1$
$A + 1 = 0$
$A = -1$
The Partial Fraction is:
$\frac{-1}{x+2} + \frac{1}{x-1}$
Checking the Result:
$\frac{-x + 1 + x + 2}{(x+2)(x-1)} = \frac{3}{(x+2)(x-1)}$
Therefore, the answer is correct.
