Answer
$\dfrac{x}{(x+1)^2}=\dfrac{1}{x+1}-\dfrac{1}{(x+1)^2}$
Work Step by Step
We are given the fraction:
$\dfrac{x}{(x+1)^2}$
As the denominator is already factored, we can write the partial fraction decomposition:
$\dfrac{x}{(x+1)^2}=\dfrac{A}{x+1}+\dfrac{B}{(x+1)^2}$
Multiply all terms by the least common denominator $(x+1)^2$:
$(x+1)^2\cdot\dfrac{x}{(x+1)^2}=(x+1)^2\cdot\dfrac{A}{x+1}+(x+1)^2\cdot\dfrac{B}{(x+1)^2}$
$x=A(x+1)+B$
$x=Ax+A+B$
$x=Ax+(A+B)$
Identify the coefficients and build the system of equations:
$\begin{cases}
A=1\\
A+B=0
\end{cases}$
Solve the system:
$A=1$
$1+B=0$
$B=-1$
The partial fraction decomposition is:
$\dfrac{x}{(x+1)^2}=\dfrac{1}{x+1}-\dfrac{1}{(x+1)^2}$
