Rational Expressions Cheat Sheet

A printable reference covering simplifying, multiplying, dividing, adding, subtracting, complex fractions, and excluded values for grades 9-11.

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Rational expressions are fractions made from polynomials, usually written as $\frac{P(x)}{Q(x)}$ with $Q(x) \ne 0$ . This topic is important because many algebra equations, functions, and word problems use rational expressions. A cheat sheet helps students remember the correct order of steps for factoring, canceling, and combining fractions. It also helps prevent mistakes with values that make a denominator equal to zero. The core skill is to factor first, then cancel only common factors, not separate terms. Excluded values come from the original denominator, so they must be found before simplifying. Multiplying and dividing rational expressions use factor cancellation, while adding and subtracting require a least common denominator. Complex fractions can usually be simplified by multiplying the numerator and denominator by the LCD.

Key Facts

A rational expression has the form $\frac{P(x)}{Q(x)}$ , where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \ne 0$ .
Excluded values are found by solving the original denominator equation $Q(x)=0$ .
To simplify, factor completely and cancel common factors, such as $\frac{(x+3)(x-2)}{(x-2)(x+5)}=\frac{x+3}{x+5}$ with $x \ne 2, -5$ .
Multiplication follows $\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$ , where $b \ne 0$ and $d \ne 0$ .
Division follows $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}=\frac{ad}{bc}$ , where $b \ne 0$ , $c \ne 0$ , and $d \ne 0$ .
Fractions with the same denominator combine as $\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$ and $\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}$ .
For unlike denominators, use the LCD, as in $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$ when $b \ne 0$ and $d \ne 0$ .
A complex fraction such as $\frac{\frac{1}{x}+2}{\frac{3}{x}}$ can be simplified by multiplying the numerator and denominator by the LCD, which is $x$ .

Vocabulary

Rational expression: A rational expression is a fraction of polynomials written as $\frac{P(x)}{Q(x)}$ , where $Q(x) \ne 0$ .
Excluded value: An excluded value is any value of the variable that makes a denominator equal to zero.
Common factor: A common factor is a factor that appears in both the numerator and denominator, such as $x+2$ in $\frac{(x+2)(x-1)}{(x+2)(x+5)}$ .
Least common denominator: The least common denominator, or LCD, is the smallest expression that contains every denominator factor needed to combine rational expressions.
Complex fraction: A complex fraction is a fraction that contains one or more smaller fractions in its numerator, denominator, or both.
Equivalent rational expressions: Equivalent rational expressions have the same value for all allowed variable values, such as $\frac{x^2-4}{x-2}$ and $x+2$ when $x \ne 2$ .

Common Mistakes to Avoid

Canceling terms instead of factors is wrong because only common factors may be canceled. In $\frac{x+2}{x+5}$ , the $x$ terms cannot be canceled because they are part of sums.
Forgetting excluded values after simplifying is wrong because restrictions come from the original expression. For example, $\frac{x^2-4}{x-2}$ simplifies to $x+2$ , but $x \ne 2$ still applies.
Adding denominators is wrong because rational expressions need a common denominator. The expression $\frac{1}{x}+\frac{1}{y}$ becomes $\frac{x+y}{xy}$ , not $\frac{2}{x+y}$ .
Dividing without multiplying by the reciprocal is wrong because division by a fraction changes to multiplication by its reciprocal. Use $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$ .
Losing a negative sign while factoring is wrong because it changes the expression. For example, $2-x=-(x-2)$ , so the negative factor must be kept.

Practice Questions

1 Evaluate $\frac{x^2-9}{x+3}$ when $x=5$ , and state the excluded value.
2 Add and simplify $\frac{2}{3x}+\frac{5}{6x}$ , including any excluded value.
3 Simplify $\frac{x^2-25}{x^2+3x-10}$ and state all excluded values.
4 Explain why a value that was canceled from the denominator is still excluded from the simplified rational expression.