Multiply and Divide Rational Expressions

By the end of this section, you will be able to: multiply rational expressions, and divide rational expressions.

Multiply rational expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

Multiplication of rational expressions. If $p$ , $q$ , $r$ , $s$ are polynomials where $q \neq 0$ and $s \neq 0$ , then

\tfrac{p}{q} \cdot \tfrac{r}{s} = \tfrac{pr}{qs}

To multiply rational expressions, multiply the numerators and multiply the denominators.

We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.

Example. Multiply $\tfrac{10}{28} \cdot \tfrac{8}{15}$ .

Multiply the numerators and denominators, look for common factors, and then remove them:

\begin{array}{lrcl} \text{Multiply the numerators and denominators.} & \tfrac{10}{28} \cdot \tfrac{8}{15} &=& \tfrac{10 \cdot 8}{28 \cdot 15} \\[10pt] \text{Factor and remove common factors.} && & \tfrac{2 \cdot 5 \cdot 2 \cdot 4}{7 \cdot 4 \cdot 3 \cdot 5} \\[10pt] \text{Simplify.} && & \tfrac{4}{21} \end{array}

Multiply: $\tfrac{6}{10} \cdot \tfrac{15}{12}$ .

Throughout this chapter, we assume that all numerical values that would make a denominator zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero.

Example. Multiply $\tfrac{2x}{3y^2} \cdot \tfrac{6xy^3}{x^2 y}$ .

Multiply the numerators and denominators, factor completely, and then remove common factors:

\begin{array}{lrcl} \text{Multiply.} & \tfrac{2x}{3y^2} \cdot \tfrac{6xy^3}{x^2 y} &=& \tfrac{2x \cdot 6xy^3}{3y^2 \cdot x^2 y} \\[10pt] \text{Factor and remove common factors.} && & \tfrac{2 \cdot x \cdot 2 \cdot 3 \cdot x \cdot y \cdot y \cdot y}{3 \cdot y \cdot y \cdot x \cdot x \cdot y} \\[10pt] \text{Simplify.} && & 4 \end{array}

Multiply: $\tfrac{3pq}{q^2} \cdot \tfrac{5p^2 q}{6pq}$ .

For rational expressions with polynomial numerators and denominators, the same steps apply — the key is to factor everything completely first, so the common factors are visible.

Example. Multiply $\tfrac{2x}{x^2 - 7x + 12} \cdot \tfrac{x^2 - 9}{6x^2}$ .

Factor the numerator and denominator of each fraction completely. Multiply the numerators and denominators (writing the monomials first is helpful), then divide out the common factors, leaving the denominator in factored form:

\begin{array}{lrcl} \text{Factor completely.} &&& \tfrac{2x}{(x - 3)(x - 4)} \cdot \tfrac{(x - 3)(x + 3)}{6x^2} \\[10pt] \text{Multiply.} && & \tfrac{2x(x - 3)(x + 3)}{6x^2(x - 3)(x - 4)} \\[10pt] \text{Divide out common factors.} && & \tfrac{x + 3}{3x(x - 4)} \end{array}

Multiply a rational expression.

Factor each numerator and denominator completely.
Multiply the numerators and denominators.
Simplify by dividing out common factors.

Example. Multiply $\tfrac{n^2 - 7n}{n^2 + 2n + 1} \cdot \tfrac{n + 1}{2n}$ .

Factor each numerator and denominator, multiply, and then remove common factors:

\begin{array}{lrcl} \text{Factor.} &&& \tfrac{n(n - 7)}{(n + 1)(n + 1)} \cdot \tfrac{n + 1}{2n} \\[10pt] \text{Multiply.} && & \tfrac{n(n - 7)(n + 1)}{(n + 1)(n + 1)2n} \\[10pt] \text{Simplify.} && & \tfrac{n - 7}{2(n + 1)} \end{array}

Multiply: $\tfrac{5x}{x^2 + 5x + 6} \cdot \tfrac{x^2 - 4}{10x}$ .

When one of the factors is the opposite of a factor in the other fraction, a factor of $-1$ appears. Remember that $a - b = -(b - a)$ , so $\tfrac{4 - x}{x - 4} = -1$ .

Example. Multiply $\tfrac{16 - 4x}{2x - 12} \cdot \tfrac{x^2 - 5x - 6}{x^2 - 16}$ .

Factor each numerator and denominator, multiply, and remove common factors. Because $4 - x$ and $x - 4$ are opposites, they divide to $-1$ :

\begin{array}{lrcl} \text{Factor.} &&& \tfrac{4(4 - x)}{2(x - 6)} \cdot \tfrac{(x - 6)(x + 1)}{(x - 4)(x + 4)} \\[10pt] \text{Multiply.} && & \tfrac{4(4 - x)(x - 6)(x + 1)}{2(x - 6)(x - 4)(x + 4)} \\[10pt] \text{Simplify (opposites divide to } -1\text{).} && & -\tfrac{2(x + 1)}{x + 4} \end{array}

Multiply: $\tfrac{12x - 6x^2}{x^2 + 8x} \cdot \tfrac{x^2 + 11x + 24}{x^2 - 4}$ . (A factor of $-1$ appears from opposite binomials.)

Divide rational expressions

To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions. The reciprocal of $\tfrac{a}{b}$ is $\tfrac{b}{a}$ — we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.

Division of rational expressions. If $p$ , $q$ , $r$ , $s$ are polynomials where $q \neq 0$ , $r \neq 0$ , $s \neq 0$ , then

\tfrac{p}{q} \div \tfrac{r}{s} = \tfrac{p}{q} \cdot \tfrac{s}{r}

To divide rational expressions, multiply the first fraction by the reciprocal of the second.

Example. Divide $\tfrac{x + 9}{6 - x} \div \tfrac{x^2 - 81}{x - 6}$ .

Rewrite the division as multiplication by the reciprocal of the second fraction, factor completely, multiply, and simplify. Remember that opposites divide to $-1$ :

\begin{array}{lrcl} \text{Multiply by the reciprocal.} &&& \tfrac{x + 9}{6 - x} \cdot \tfrac{x - 6}{x^2 - 81} \\[10pt] \text{Factor.} && & \tfrac{x + 9}{6 - x} \cdot \tfrac{x - 6}{(x - 9)(x + 9)} \\[10pt] \text{Multiply.} && & \tfrac{(x + 9)(x - 6)}{(6 - x)(x - 9)(x + 9)} \\[10pt] \text{Simplify (opposites divide to } -1\text{).} && & -\tfrac{1}{x - 9} \end{array}

Divide rational expressions.

Rewrite the division as the product of the first rational expression and the reciprocal of the second.
Factor the numerators and denominators completely.
Multiply the numerators and denominators together.
Simplify by dividing out common factors.

Example. Divide $\tfrac{3n^2}{n^2 - 4n} \div \tfrac{9n^2 - 45n}{n^2 - 7n + 10}$ .

Rewrite as multiplication by the reciprocal, factor everything, and then divide out common factors:

\begin{array}{lrcl} \text{Multiply by the reciprocal.} &&& \tfrac{3n^2}{n^2 - 4n} \cdot \tfrac{n^2 - 7n + 10}{9n^2 - 45n} \\[10pt] \text{Factor and multiply.} && & \tfrac{3 \cdot n \cdot n \cdot (n - 5)(n - 2)}{n(n - 4) \cdot 3 \cdot 3 \cdot n \cdot (n - 5)} \\[10pt] \text{Simplify.} && & \tfrac{n - 2}{3(n - 4)} \end{array}

Divide: $\tfrac{c + 3}{5 - c} \div \tfrac{c^2 - 9}{c - 5}$ . (A factor of $-1$ appears from opposite binomials.)

Divide: $\tfrac{2m^2}{m^2 - 8m} \div \tfrac{8m^2 + 24m}{m^2 + m - 6}$ .

Sometimes we divide a rational expression by a polynomial. Remember that a fraction bar means division, and any polynomial can be written as a fraction over $1$ . To divide a fraction by a whole number, we first write the whole number as a fraction so we can find its reciprocal — for example, $\tfrac{3}{5} \div 4 = \tfrac{3}{5} \div \tfrac{4}{1} = \tfrac{3}{5} \cdot \tfrac{1}{4}$ .

Example. Divide $\tfrac{a^2 - b^2}{3ab} \div (a^2 + 2ab + b^2)$ .

Write the second expression as a fraction over $1$ , multiply by its reciprocal, factor, and simplify:

\begin{array}{lrcl} \text{Write as a fraction over } 1. &&& \tfrac{a^2 - b^2}{3ab} \div \tfrac{a^2 + 2ab + b^2}{1} \\[10pt] \text{Multiply by the reciprocal.} && & \tfrac{a^2 - b^2}{3ab} \cdot \tfrac{1}{a^2 + 2ab + b^2} \\[10pt] \text{Factor and multiply.} && & \tfrac{(a - b)(a + b) \cdot 1}{3ab \cdot (a + b)(a + b)} \\[10pt] \text{Simplify.} && & \tfrac{a - b}{3ab(a + b)} \end{array}

Divide: $\tfrac{2x^2 - 14x - 16}{4} \div (x^2 + 2x + 1)$ .

A complex fraction is another way of writing division of two fractions — the fraction bar means “divide the top by the bottom.”

Example. Divide the complex fraction

\cfrac{\frac{6x^2 - 7x + 2}{4x - 8}}{\frac{2x^2 - 7x + 3}{x^2 - 5x + 6}}.

Rewrite the complex fraction with a division sign, then as the product of the first fraction and the reciprocal of the second. Factor everything and divide out common factors:

\begin{array}{lrcl} \text{Rewrite as division.} &&& \tfrac{6x^2 - 7x + 2}{4x - 8} \div \tfrac{2x^2 - 7x + 3}{x^2 - 5x + 6} \\[10pt] \text{Multiply by the reciprocal.} && & \tfrac{6x^2 - 7x + 2}{4x - 8} \cdot \tfrac{x^2 - 5x + 6}{2x^2 - 7x + 3} \\[10pt] \text{Factor and multiply.} && & \tfrac{(2x - 1)(3x - 2)(x - 2)(x - 3)}{4(x - 2)(2x - 1)(x - 3)} \\[10pt] \text{Simplify.} && & \tfrac{3x - 2}{4} \end{array}

If we have more than two rational expressions to work with, we follow the same procedure: rewrite any division as multiplication by the reciprocal, then factor and multiply.

Example. Perform the indicated operations: $\tfrac{3x - 6}{4x - 4} \cdot \tfrac{x^2 + 2x - 3}{x^2 - 3x - 10} \div \tfrac{2x + 12}{8x + 16}$ .

Rewrite the division as multiplication by the reciprocal, factor completely, multiply, and divide out common factors:

\begin{array}{lrcl} \text{Multiply by the reciprocal.} &&& \tfrac{3x - 6}{4x - 4} \cdot \tfrac{x^2 + 2x - 3}{x^2 - 3x - 10} \cdot \tfrac{8x + 16}{2x + 12} \\[10pt] \text{Factor and multiply.} && & \tfrac{3 \cdot 8(x - 2)(x + 3)(x - 1)(x + 2)}{4 \cdot 2(x - 1)(x + 2)(x - 5)(x + 6)} \\[10pt] \text{Simplify.} && & \tfrac{3(x - 2)(x + 3)}{(x - 5)(x + 6)} \end{array}

Perform the indicated operations: $\tfrac{4m + 4}{3m - 15} \cdot \tfrac{m^2 - 3m - 10}{m^2 - 4m - 32} \div \tfrac{12m - 36}{6m - 48}$ .

Key terms

rational expression — a fraction whose numerator and denominator are polynomials. reciprocal — the fraction obtained by interchanging the numerator and denominator; the reciprocal of $\tfrac{a}{b}$ is $\tfrac{b}{a}$ . complex fraction — a fraction in which the numerator, the denominator, or both contain a fraction; a complex fraction represents the division of its top by its bottom.

This section is adapted from Elementary Algebra 2e, Section 8.2: Multiply and Divide Rational Expressions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the worked “How To” step tables as display equality chains with left-hand explanations, and stated each simplification as a divide-out of common factors; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

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