Algebraic fraction

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.

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