# Algebraic fraction

sort of mathematical expression

An algebraic fraction is a fraction where the top and the bottom are algebraic expressions. Two examples of algebraic fractions are ${\displaystyle {\frac {3x}{x^{2}+2x-3}}}$ and ${\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}$.

A rational fraction is an algebraic fraction where the top and the bottom are polynomials. ${\displaystyle {\frac {3x}{x^{2}+2x-3}}}$ is a rational fraction, but ${\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}$ is not. This is because the top has a square root function.

## Operations

### Multiplication

${\displaystyle {\frac {x^{2}+9x+20}{x^{2}-4}}\cdot {\frac {x+2}{x+4}}}$

${\displaystyle ={\frac {{\color {Red}(x+4)}(x+5)}{{\color {Red}(x+2)}(x-2)}}\cdot {\frac {\color {Red}x+2}{\color {Red}x+4}}}$

${\displaystyle ={\frac {x+5}{x-2}}}$

### Division

Turn the equation into multiplication by flipping one of the fractions. After that, do what is shown above.

${\displaystyle {\frac {x^{2}+9x+20}{x^{2}-4}}{\color {Green}\div }{\frac {\color {Orange}x+2}{\color {Blue}x+4}}={\frac {x^{2}+9x+20}{x^{2}-4}}{\color {Green}\cdot }{\frac {\color {Blue}x+4}{\color {Orange}x+2}}}$