Algebraic fraction

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

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

Operations

Multiplication

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

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

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

Division

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

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