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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

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*}

Review (Answers)

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

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Vocabulary

Rational Expression

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

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