6 Partial Fractions

Polynomial fractions of the form $\frac{ax+b}{cx^2+dx+e} $ can often be simplified by means of partial fractions. If the denominator can be factorised then the fraction can be split into the sum of simpler fractions.

Example 6.1: Simplify the following expression $\frac{5x+1}{x^2+x-2} $

Start the process of simplification by factorising the denominator.
$x^2+x-2$$=$$(x-1)(x+2)$
Next, imagine there exists $A$ and $B$ such that
$\frac{5x+1}{x^2+x-2} $$=$$\frac{A}{x-1}+\frac{B}{x+2} $
Put the right hand side over a common denominator
$\frac{5x+1}{x^2+x-2} $$=$$\frac{A(x+2)}{(x-1)(x+2)}+\frac{B(x-1)}{(x-1)(x+2)} $
which means    $ 5x+1 $$=$$A(x+2)+B(x-1) $
If we let $x=1$ we can see $A=2$. If we let $x=-2$ $B=3$.
so we have    $\frac{5x+1}{x^2+x-2} $$=$$\frac{2}{x-1}+\frac{3}{x+2} $

If a fraction has repeated factors in the denominator we need partial fractions that account for all the possible factors. To do this we put $A$ over the factor, $B$ over the square of the factor and so on until we have all the possible factors. For example, an expression of the form $\frac{x+a}{(x+b)^2}$ would be factorised like this $\frac{A}{(x+b)}+\frac{B}{(x+b)^2}$.

$\frac{x+a}{(x+b)^2}=\frac{A}{(x+b)}+\frac{B}{(x+b)^2}$.

If a fraction contains non-linear factor of the form $ax^2+bx+c$ we need a numerator that includes terms in $x$ that up to an order one less than the factor in the denominator. The partial fraction for a factor of the form $ax^2+bx+c$ would be $\frac{Ax+B}{ax^2+bx+c}$.

$\frac{x+a}{ax^2+bx+c}=\frac{Ax+B}{ax^2+bx+c}$.

In cases where the order of the numerator is the same as the order of the denominator extract a constant term and then find partial fractions for the remainder. For example if we have the fraction $\frac{x^2-3x+2}{x^2+2x+1}$ the numerator can be rewritten as $x^2+2x+1+(-5x+1)$ which means the fraction can be written as $\frac{x^2-3x+2}{x^2+2x+1}=1+\frac{(-5x+1)}{x^2+2x+1}$ which can be solved with the

In cases where the order of the numerator is higher than the order of the denominator we divide the numerator by the denominator which will give a function of $x$. You can then find partial freactions for the remainder.