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Partial fraction expansion, also know as Fractional Decomposition, is a method of expanding an algebraic fraction into the sum of several expressions.

## Procedure

For partial fraction expansion to be possible, the denominator must be of a higher degree than the numerator. If it is, it must be factored. For example:

$\frac{x+2}{x^2+5x+4}=\frac{x+2}{(x+4)(x+1)}$

Now we can make the numerators equal to $A,B$ and cross multiplying.

$\frac{x+2}{(x+4)(x+1)}=\frac{A}{(x+4)}+\frac{B}{(x+1)}=\frac{A(x+1)+B(x+4)}{(x+4)(x+1)}$

We can now multiply by the denominator.

$x+2=A(x+1)+B(x+4)$

$A,B$ can be solved for setting $x$ equal to a number that will cancel one term out.

\begin{align}&x=-4 \\&(-4)+2=A\big((-4)+1\big)+B\big((-4)+4\big) \\&-2=A(-3)+B(0)=-3A \\&-2=-3A \\&A=\frac23 \\\\&x=-1 \\&(-1)+2=A\big((-1)+1\big)+B\big((-1)+4\big) \\&1=A(0)+B(3)=3B \\&1=3B \\&B=\frac13\end{align}

We now know that

$\frac{x+2}{(x+4)(x+1)}=\frac{\frac{2}{3}}{(x+4)}+\frac{\frac{1}{3}}{(x+1)}=\frac{2}{3(x+4)}+\frac{1}{3(x+1)}$

which can be proved by cross multiplication.

## Applications

The main use of partial fraction expansion is in integral calculus, where it can be used to find an antiderivative. Since we know that

$\int\big(f(x)+g(x)\big)dx=\int f(x)dx+\int g(x)dx$

we can use partial fraction expansion to solve seemingly difficult integrals. For example, we just found that

$\frac{x+2}{(x+4)(x+1)}=\frac{2}{3(x+4)}+\frac{1}{3(x+1)}$

This means that

\begin{align}\int\dfrac{x+2}{(x+4)(x+1)}dx&=\int\left(\frac{2}{3(x+4)}+\frac{1}{3(x+1)}\right)dx\\&=\int\frac{2}{3(x+4)}dx+\int\frac{dx}{3(x+1)}\end{align}

This can easily be solved to find that

$\int\dfrac{x+2}{(x+4)(x+1)}dx=\frac{2\ln\big(|x+4|\big)+\ln\big(|x+1|\big)}{3}+C$

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