Algebraic fraction

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

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

Multiplication

[math]\displaystyle{ \frac{x^2+9x+20}{x^2-4} \cdot \frac{x+2}{x+4} }[/math]

[math]\displaystyle{ = \frac{{\color{Red}(x+4)}(x+5)}{{\color{Red}(x+2)}(x-2)} \cdot \frac{\color{Red}x+2}{\color{Red}x+4} }[/math]

[math]\displaystyle{ = \frac{x+5}{x-2} }[/math]

Division

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

[math]\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} }[/math]