Numerical Fractions & Algebraic Fractions (Rational Expressions)
Big Idea: Just as numerical fractions represent the ratio of two numbers, algebraic fractions (also called rational expressions) represent the ratio of two algebraic expressions. Mastering algebraic fractions unlocks the door to solving complex equations, modeling real-world situations, and preparing for advanced algebra.
Numerical Fraction (Review)
A numerical fraction is a ratio of two integers:
$\dfrac{a}{b}$ where $a, b \in \mathbb{Z}$ and $b \neq 0$
Examples:
- $\dfrac{3}{5}$, $\dfrac{-7}{12}$, $\dfrac{15}{4}$
We simplify by cancelling common numerical factors:
$\dfrac{12}{18} = \dfrac{6 \times 2}{6 \times 3} = \dfrac{2}{3}$
Algebraic Fraction (Our Focus)
An algebraic fraction (rational expression) is a ratio where the numerator and/or denominator contain variables:
$\dfrac{P(x)}{Q(x)}$ where $Q(x) \neq 0$
Examples:
- $\dfrac{x+2}{x-3}$, $\dfrac{2a^2+5a}{a+1}$, $\dfrac{3x-1}{4x}$
We simplify by cancelling common algebraic factors:
$\dfrac{x^2 - 4}{x + 2} = \dfrac{(x-2)(x+2)}{(x+2)} = x - 2$