I need help with these...
Partial Fraction Decomposition:
1/4x^2-9
and
-2x+15/x^2-x-12
Hi RenSully,
Here's the second one:
If all the factors are of the form ( x - c ), then each of the partial fractions will be of the form $\displaystyle \frac{A}{x-c}$.
$\displaystyle \frac{-2x+15}{x^2-x-12}=\frac{-2x+15}{(x-4)(x+3)}\Rightarrow\frac{A}{x-4}+\frac{B}{x+3}$
First, multiply the original proper fraction by ( x - c ) and evaluate the result at x = c to get the value of the constant.
$\displaystyle \frac{-2x+15}{(x-4)(x+3)}(x-4)=\frac{-2x+15}{x+3}\Rightarrow A=\frac{-2(4)+15}{4+3}=1$
$\displaystyle \frac{-2x+15}{(x-4)(x+3)}(x+3)=\frac{-2x+15}{x-4}\Rightarrow B=\frac{-2(-3)+15}{-3-4}=-3$
Thus:
$\displaystyle \frac{-2x+15}{x^2-x-12}=\frac{1}{x-4}+\frac{-3}{x+3}$