# How to simplify algebraic fractions?

Mar 22, 2018

There is no single method for simplifying algebraic fractions.

#### Explanation:

Some algebraic fractions can not be simplified.
For example: $\frac{x}{y + 1}$ can not be simplified.

Sometimes a common factor can be found in the numerator and denominator and removed.
For example: $\frac{{x}^{2} - 1}{x y - 2 x - y - 2} = \frac{\left(x + 1\right) \left(x - 1\right)}{\left(x - 1\right) \left(y + 2\right)} = \frac{x + 1}{y + 2}$

Sometimes you can perform polynomial long division and find a simplification; but sometimes whether the result is a "simplification" is questionable:
For example: $\frac{2 {x}^{3} + 4 {x}^{2} + 9}{x - 3}$
if treated as a long division gives the result:
$\textcolor{w h i t e}{\text{XXX}} 2 {x}^{2} + 5 x + 15 + \frac{54}{x - 3}$
Is this really a simplification?