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Multiplication of Rational Expressions

Multiply and reduce fractions with variables in the denominator

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Multiplication of Rational Expressions

What if you had two rational expressions like \begin{align*}\frac{2x^2 - 3}{x - 4}\end{align*}2x23x4 and \begin{align*}\frac{x^2 - 3x + 2}{x^2}\end{align*}x23x+2x2 and you wanted to multiply them? How could you do so such that the answer were in simplest terms? After completing this Concept, you'll be able to multiply rational expressions like this one.

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CK-12 Foundation: 1208S Multiplying 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*}abcd=acbd

Multiply Rational Expressions Involving Monomials

Example A

Multiply the following: \begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2}\end{align*}a16b84b35a2.


Cancel common factors from the numerator and denominator. The common factors are 4, \begin{align*}a\end{align*}a, and \begin{align*}b^3\end{align*}b3. Canceling them out leaves \begin{align*}\frac{1}{4b^5} \cdot \frac{1}{5a} = \frac{1}{20ab^5}\end{align*}14b515a=120ab5.

Example B

Multiply \begin{align*}9x^2 \cdot \frac{4y^2}{21x^4}\end{align*}9x24y221x4.


Rewrite the problem as a product of two fractions: \begin{align*}\frac{9x^2}{1} \cdot \frac{4y^2}{21x^4}\end{align*}9x214y221x4 Then cancel common factors from the numerator and denominator.

The common factors are 3 and \begin{align*}x^2\end{align*}x2. Canceling them out leaves \begin{align*}\frac{3}{1} \cdot \frac{4y^2}{7x^2} = \frac{12y^2}{7x^2}\end{align*}314y27x2=12y27x2.

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*}4x+123x2xx29.


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*}4(x+3)3x2x(x+3)(x3)

The common factors are \begin{align*}x\end{align*}x and \begin{align*}(x + 3)\end{align*}(x+3). 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*}43x1(x3)=43x(x3)=43x29x.

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*}3x+184x2+19x5(x2+3x10).


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*}3x+184x2+19x5x2+3x101

Factor polynomials: \begin{align*}\frac{3(x+6)}{(x+5)(4x-1)} \cdot \frac{(x-2)(x+5)}{1}\end{align*}3(x+6)(x+5)(4x1)(x2)(x+5)1

The common factor is \begin{align*}(x + 5)\end{align*}(x+5). 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*}3(x+6)(4x1)(x2)1=3(x+6)(x2)(4x1)=3x2+12x364x1

Watch this video for help with the Examples above.

CK-12 Foundation: Multiplying Rational Expressions

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*}12x2x6x21x2+7x+64x227x+18.


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*}(3x+2)(4x3)(x+1)(x1)(x+1)(x+6)(4x3)(x6).

The common factors are \begin{align*}(x + 1)\end{align*}(x+1) and \begin{align*}(4x - 3)\end{align*}(4x3). 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*}(3x+2)(x1)(x+6)(x6)=(3x+2)(x+6)(x1)(x6)=3x2+20x+12x27x+6

Explore More

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*}x32y32y2x
  2. \begin{align*}\frac{2x}{y^2} \cdot \frac{4y}{5x}\end{align*}2xy24y5x
  3. \begin{align*}2xy \cdot \frac{2y^2}{x^3}\end{align*}2xy2y2x3
  4. \begin{align*}\frac{4y^2-1}{y^2-9} \cdot \frac{y-3}{2y-1}\end{align*}4y21y29y32y1
  5. \begin{align*}\frac{6ab}{a^2} \cdot \frac{a^3b}{3b^2}\end{align*}6aba2a3b3b2
  6. \begin{align*}\frac{33a^2}{-5} \cdot \frac{20}{11a^3}\end{align*}33a252011a3
  7. \begin{align*}\frac{2x^2+2x-24}{x^2+3x} \cdot \frac{x^2+x-6}{x+4}\end{align*}2x2+2x24x2+3xx2+x6x+4
  8. \begin{align*}\frac{x}{x-5} \cdot \frac{x^2-8x+15}{x^2-3x}\end{align*}xx5x28x+15x23x
  9. \begin{align*}\frac{5x^2+16x+3}{36x^2-25} \cdot (6x^2+5x)\end{align*}5x2+16x+336x225(6x2+5x)
  10. \begin{align*}\frac{x^2+7x+10}{x^2-9} \cdot \frac{x^2-3x}{3x^2+4x-4}\end{align*}x2+7x+10x29x23x3x2+4x4
  11. \begin{align*}\frac{x^2+8x+16}{7x^2+9x+2} \cdot \frac{7x+2}{x^2+4x}\end{align*}x2+8x+167x2+9x+27x+2x2+4x

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 12.8. 


Rational Expression

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

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