<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 12.8: Multiplication of Rational Expressions

Difficulty Level: At Grade Created by: CK-12
Estimated19 minsto complete
%
Progress
Practice Multiplication of Rational Expressions

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated19 minsto complete
%
Estimated19 minsto complete
%
MEMORY METER
This indicates how strong in your memory this concept is

### Multiplication of Rational Expressions

The rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing rational numbers, so let’s start by reviewing multiplication and division of fractions. When we multiply two fractions we multiply the numerators and denominators separately:

\begin{align*}\frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d}\end{align*}

#### Multiplying Rational Expressions Involving Monomials

1. Multiply the following: \begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2}\end{align*}.

Cancel common factors from the numerator and denominator. The common factors are 4, \begin{align*}a\end{align*}, and \begin{align*}b^3\end{align*}. Canceling them out leaves \begin{align*}\frac{1}{4b^5} \cdot \frac{1}{5a} = \frac{1}{20ab^5}\end{align*}.

2. Multiply \begin{align*}9x^2 \cdot \frac{4y^2}{21x^4}\end{align*}.

Rewrite the problem as a product of two fractions: \begin{align*}\frac{9x^2}{1} \cdot \frac{4y^2}{21x^4}\end{align*} Then cancel common factors from the numerator and denominator.

The common factors are 3 and \begin{align*}x^2\end{align*}. Canceling them out leaves \begin{align*}\frac{3}{1} \cdot \frac{4y^2}{7x^2} = \frac{12y^2}{7x^2}\end{align*}.

#### Multiplying Rational Expressions Involving Polynomials

When multiplying rational expressions involving polynomials, first we need to factor all polynomial expressions as much as we can. Then we follow the same procedure as before.

Multiply \begin{align*}\frac{4x+12}{3x^2} \cdot \frac{x}{x^2-9}\end{align*}.

Factor all polynomial expressions as much as possible: \begin{align*}\frac{4(x+3)}{3x^2} \cdot \frac{x}{(x+3)(x-3)}\end{align*}

The common factors are \begin{align*}x\end{align*} and \begin{align*}(x + 3)\end{align*}. Canceling them leaves \begin{align*}\frac{4}{3x} \cdot \frac{1}{(x-3)} = \frac{4}{3x(x-3)} = \frac{4}{3x^2-9x}\end{align*}.

#### Multiplying a Rational Expression by a Polynomial

When we multiply a rational expression by a whole number or a polynomial, we can write the whole number (or polynomial) as a fraction with denominator equal to one. We then proceed the same way as in the previous examples.

Multiply \begin{align*}\frac{3x+18}{4x^2+19x-5} \cdot (x^2+3x-10)\end{align*}.

Rewrite the expression as a product of fractions: \begin{align*}\frac{3x+18}{4x^2+19x-5} \cdot \frac{x^2+3x-10}{1}\end{align*}

Factor polynomials: \begin{align*}\frac{3(x+6)}{(x+5)(4x-1)} \cdot \frac{(x-2)(x+5)}{1}\end{align*}

The common factor is \begin{align*}(x + 5)\end{align*}. Canceling it leaves \begin{align*}\frac{3(x+6)}{(4x-1)} \cdot \frac{(x-2)}{1} = \frac{3(x+6)(x-2)}{(4x-1)} = \frac{3x^2+12x-36}{4x-1}\end{align*}

### Example

#### Example 1

Multiply \begin{align*}\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18}\end{align*}.

Factor polynomials: \begin{align*}\frac{(3x+2)(4x-3)}{(x+1)(x-1)} \cdot \frac{(x+1)(x+6)}{(4x-3)(x-6)}\end{align*}.

The common factors are \begin{align*}(x + 1)\end{align*} and \begin{align*}(4x - 3)\end{align*}. Canceling them leaves \begin{align*}\frac{(3x+2)}{(x-1)} \cdot \frac{(x+6)}{(x-6)} = \frac{(3x+2)(x+6)}{(x-1)(x-6)} = \frac{3x^2+20x+12}{x^2-7x+6}\end{align*}

### Review

Multiply the following rational expressions and reduce the answer to lowest terms.

1. \begin{align*}\frac{x^3}{2y^3} \cdot \frac{2y^2}{x}\end{align*}
2. \begin{align*}\frac{2x}{y^2} \cdot \frac{4y}{5x}\end{align*}
3. \begin{align*}2xy \cdot \frac{2y^2}{x^3}\end{align*}
4. \begin{align*}\frac{4y^2-1}{y^2-9} \cdot \frac{y-3}{2y-1}\end{align*}
5. \begin{align*}\frac{6ab}{a^2} \cdot \frac{a^3b}{3b^2}\end{align*}
6. \begin{align*}\frac{33a^2}{-5} \cdot \frac{20}{11a^3}\end{align*}
7. \begin{align*}\frac{2x^2+2x-24}{x^2+3x} \cdot \frac{x^2+x-6}{x+4}\end{align*}
8. \begin{align*}\frac{x}{x-5} \cdot \frac{x^2-8x+15}{x^2-3x}\end{align*}
9. \begin{align*}\frac{5x^2+16x+3}{36x^2-25} \cdot (6x^2+5x)\end{align*}
10. \begin{align*}\frac{x^2+7x+10}{x^2-9} \cdot \frac{x^2-3x}{3x^2+4x-4}\end{align*}
11. \begin{align*}\frac{x^2+8x+16}{7x^2+9x+2} \cdot \frac{7x+2}{x^2+4x}\end{align*}

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Rational Expression A rational expression is a fraction with polynomials in the numerator and the denominator.

Show Hide Details
Description
Difficulty Level:
Authors:
Tags:
Subjects: