# Partial Fractions

In this article, we will be learning about partial fractions for proper fractions. More specifically, we will be concentrating on the first two cases listed below.

• Case 1:

Partial fractions with distinct linear factors in the denominator.

• Case 2:

Proper fractions with a repeated linear factor in the denominator.

• Case 3:

Proper fractions with a quadratic factor that cannot be factorised in the denominator.

## What are Partial Fractions?

A proper algebraic fraction may be expressed in two or more partial fractions.

The process from left to right shows the result of subtracting two algebraic fractions.

 $\begin{gather} \xrightarrow {\text {Simplification of Algebraic Fractions}} \end{gather}$ \begin{align*} \frac{3}{x+1} -\frac{5}{2x-3}=\frac{x-14}{(x+1)(2x-3)} \end{align*} $\begin{gather} \xleftarrow {\text {Partial Fractions}} \end{gather}$

The reverse process from right to left shows the ‘splitting’ of an algebraic fraction into two or more partial fractions.

## Case 1: Proper Fractions with Distinct Linear Factors in Denominator

An algebraic fraction where the denominator is a product of two distinct linear factors may be expressed in partial fractions as shown below.

\begin{align*} \frac{px+q}{(ax+b)(cx+d)}=\frac{A}{ax+b}+\frac{B}{cx+d} \end{align*}, where $A$ and $B$ are constants.

The denominator is a product of $(ax+b)$ and $(cx+d)$. $(ax+b)$ is one linear factor and $(cx+d)$ is another linear factor.

Question 1:

Express \begin{align*} \frac{(x-14)}{(x+1)(2x-3)} \end{align*} in partial fractions.

Solution:

• Let \begin{align*} \frac{(x-14)}{(x+1)(2x-3)}=\frac{A}{(x+1)}+\frac{B}{(2x-3)} \end{align*}.

We need to find the value of $A$ and of $B$

• Multiply the equation throughout by the denominator $(x + 1)(2x – 3)$

\begin{align*} \frac{(x-14)}{(x+1)(2x+3)} \times {(x+1)(2x-3)} \;=\; \frac{A}{(x+1)} \times {(x+1)(2x-3)} \;+\; \frac {B}{(2x-3)} \times {(x+1)(2x-3)} \end{align*}

• We will be left with the following equation,

\begin{align*} (x-14)=A(2x-3)+B(x+1) \end{align*}

• Then, we need to perform smart substitution. Substitute $x=-1$, so that the$B(x + 1)$ term is eliminated.

So, $x = −1$

\begin{align*} & −1−14 = A(2(−1)−3) \\ & −15 = A(−5) \\ & A = \frac{-15}{-5 } \\ &= 3 \end{align*}

• Then, substitute $x = 1.5$ to eliminate the $A(2x – 3)$ term.

\begin{align} &1.5 − 14 = A(2\times 1.5 − 3) + B(1.5 + 1) \\ &−12.5 = 2.5B \\ & B = −5 \end{align}

• Lastly, substitute the value of $A$ and of $B$ into the first equation.

Therefore, \begin{align*} \frac{(x-14)}{(x+1)(2x+3)}=\frac {3}{(x+1)}-\frac{5}{(2x-3)} \end{align*}.

## Case 2: Proper Fractions with Repeated Linear Factors in the Denominator

An algebraic fraction where the denominator contains a linear factor that is squared may be expressed in partial fractions as shown below.

\begin{align*} \frac {px+q}{(ax+b)^2}=\frac{A}{ax+b}+\frac{B}{(ax+b)^2} \end{align*}, where $A$ and $B$ are constants.

Question 2:

Express \begin{align*} \frac {x}{(x-1)(x+4)^2} \end{align*} in partial fractions.

Solution:

• Let \begin{align*} \frac {x}{(x-1)(x+4)^2}=\frac{A}{(x-1)}+\frac{B}{(x+4)}+\frac{C}{(x+4)^2} \end{align*}

• Multiply the equation throughout by the denominator $(x – 1)(x + 4)^2$

$x = A(x+4) + B(x−1)(x+4) + C(x−1)$

• Substitute $x = −4$,

\begin{align*} &−4 = C(−4−1) \\ &−4 = −5C \\ & C = \frac{4}{5} \end{align*}

• Substitute $x = 1$,

\begin{align*} & 1 = A(1+4)^2+ 0 + 0 \\ &1 = 25A \\ & A = \frac{1}{25} \end{align*}

• For the third substitution, choose any number (or a number that makes the equation easy to manipulate).

Substitute\begin{align*} x = 0, \;C = \frac{4}{5}, \;A = \frac {1}{25}, \end{align*}

\begin{align*} & 0 = \frac{1}{25}\times(0+4)^2 + B(0-1)(0+4) + \frac{4}{5} \times (0-1) \\ & 0 =\frac{16}{25} + B(−4) − \frac{4}{5} \\ & \frac{4}{5} − \frac{16}{25} = − 4B \\ & \frac{4}{25} = − 4B \\ & B = − \frac{1}{25} \end{align*}

• Lastly, substitute the values of $A$, $B$ and $C$ into the first equation.

\begin{align*} \frac{x}{(x-1)(x+4)^2}=\frac{1}{25(x-1)}-\frac{1}{25(x+4)}+\frac{4}{5(x+4)^2} \end{align*}

## Conclusion

In this article, we have covered two cases for splitting an algebraic fraction into partial fractions:

1. A proper algebraic fraction with distinct linear factors

\begin{align*} \frac{px+q}{(ax+b)(cx+d)}=\frac{A}{ax+b}+\frac{B}{cx+d} \end{align*}

1. A proper fraction with repeated linear factors

\begin{align*} \frac {px+q}{(ax+b)^2}=\frac{A}{ax+b}+\frac{B}{(ax+b)^2} \end{align*}

The only difference between these two forms is a square in the denominator. However, please take note of it as the subsequent steps will differ quite a bit!

Keep Learning! Keep Improving!

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