# Rewrite the expression f(x) = \frac {3x-5}{x^2(x-2)} using partial fraction.

## Question:

Rewrite the expression {eq}f(x) = \frac {3x-5}{x^2(x-2)} {/eq} using partial fraction.

## Partial Fraction:

A partial fraction is a mathematical way of decomposing the difficult fraction into two or more fractions.

In partial fraction method, first, we factor the denominator as much as possible and then we write one partial fraction for each factor. Next, we multiply through by the bottom so that we are no longer left with any fraction and then we solve the values of constants.

We have to rewrite the expression {eq}f(x) = \dfrac {3x-5}{x^2(x-2)} {/eq} using a partial fraction.

Applying the partial fraction method to split the fraction into a simpler form:

{eq}\begin{align} \dfrac {3x-5}{x^2(x-2)} &=\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{x-2}\\ 3x-5 &=Ax(x-2)+B(x-2)+Cx^2 \end{align} {/eq}

Let {eq}x=0 {/eq}

{eq}\begin{align} 3(0)-5 &=A(0)+B(0-2)+C(0)\\ -5 &=-2B\\ B &= \dfrac{5}{2} \end{align} {/eq}

Let {eq}x-2=0 \Rightarrow x=2 {/eq}

{eq}\begin{align} 3(2)-5 &=A(0)+B(0)+C(2)^2\\ 1 &=4C\\ C &=\dfrac{1}{4} \end{align} {/eq}

Comparing the coefficient of {eq}x^2 {/eq}

{eq}\begin{align} 0 &=A+C\\ A &=-C\\ A &=\dfrac{-1}{4} \end{align} {/eq}

Applying these values, we have:

{eq}\dfrac{3x-5}{x^2(x-2)}=\dfrac{\dfrac{-1}{4}}{x}+\dfrac{\dfrac{5}{2}}{x^2}+\dfrac{\dfrac{1}{4}}{x-2}\\ \color{blue}{\boxed{\dfrac {3x-5}{x^2(x-2)}=\dfrac{-1}{4x}+\dfrac{5}{2x^2}+\dfrac{1}{4(x-2)}}} {/eq}