What if you had two rational expressions like \begin{align*}\frac{2x^2 - 3}{x - 4}\end{align*} and \begin{align*}\frac{x^2 - 3x + 2}{x^2}\end{align*} and you wanted to multiply them? How could you do so such that the answer were in simplest terms? After completig this Concept, you'll be able to multiply rational expressions like this one.

### Watch This

CK-12 Foundation: 1208S Multiplying Rational Expressions

### Guidance

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*}

**Multiply Rational Expressions Involving Monomials**

#### Example A

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

**Solution**

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*}.

#### Example B

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

**Solution**

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*}.

**Multiply 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.

#### Example C

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

**Solution**

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*}.

**Multiply 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.

#### Example D

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

**Solution**

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*}

Watch this video for help with the Examples above.

CK-12 Foundation: Multiplying Rational Expressions

### Vocabulary

- 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*}

### Guided Practice

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

**Solution**

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*}

### Practice

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

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