Answer
$\dfrac{6x-11}{(x-1)^2}=\dfrac{6}{x-1}-\dfrac{5}{(x-1)^2}$
Work Step by Step
We are given the fraction:
$\dfrac{6x-11}{(x-1)^2}$
As the denominator is already factored, we can write the partial fraction decomposition:
$\dfrac{6x-11}{(x-1)^2}=\dfrac{A}{x-1}+\dfrac{B}{(x-1)^2}$
Multiply all terms by the least common denominator $(x-1)^2$:
$(x-1)^2\cdot\dfrac{6x-11}{(x-1)^2}=(x-1)^2\cdot\dfrac{A}{x-1}+(x-1)^2\cdot\dfrac{B}{(x-1)^2}$
$6x-11=A(x-1)+B$
$6x-11=Ax-A+B$
$6x-11=Ax+(-A+B)$
Identify the coefficients and build the system of equations:
$\begin{cases}
A=6\
-A+B=-11
\end{cases}$
Solve the system:
$A=6$
$-6+B=-11$
$B=-5$
The partial fraction decomposition is:
$\dfrac{6x-11}{(x-1)^2}=\dfrac{6}{x-1}-\dfrac{5}{(x-1)^2}$
