Example

Express $\dfrac {x^2+x+1}{x^2-2x-3}$ as the sum of partial fractions.

Solution

Begin by factorising the denominator:

Now observe that the highest degree of both numerator and denominator is the same. This means that the partial fraction decomposition will look like

We first need to work out the whole number, $A$, as follows.

Now:

Equating numerators:

Since this is true for all values of $x$, solve for $B$ and $C$ by evaluating at any two values of $x$. Choose $x=-1$ and $x=3$.

$x=-1:\qquad 1 = -4C \qquad \implies C = -\frac{1}{4}$
$x=3:\qquad 13 = 4B \qquad \implies B = \frac{13}{4}$

Thus