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Mathematics LibreTexts

8.2: Multiply and Divide Rational Expressions

  • Page ID
    18978
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    By the end of this section, you will be able to:
    • Multiply rational expressions
    • Divide rational expressions

     

    Before you get started, take this readiness quiz.

    If you miss a problem, go back to the section listed and review the material.

    1. Multiply: \(\frac{14}{15}·\frac{6}{35}\).
      If you missed this problem, review [link].
    2. Divide: \(\frac{14}{15}÷\frac{6}{35}\).
      If you missed this problem, review [link].
    3. Factor completely: \(2x^2−98\).
      If you missed this problem, review [link].
    4. Factor completely: \(10n^3+10\).
      If you missed this problem, review [link].
    5. Factor completely: \(10p^2−25pq−15q^2\).
      If you missed this problem, review [link].

    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.

    Definition: MULTIPLICATION OF RATIONAL EXPRESSIONS

    If p,q,r,s are polynomials where \(q \ne 0\) and \(s \ne 0\)

    \(\frac{p}{q}·\frac{r}{s}=\frac{pr}{qs}\)

    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 \(\PageIndex{1}\)

    Multiply: \(\frac{10}{28}·\frac{8}{15}\).

    Answer
      .
    Multiply the numerators and denominators. .
    Look for common factors, and then remove them. .
    Simplify. .

    Example \(\PageIndex{2}\)

    Mulitply: \(\frac{6}{10}·\frac{15}{12}\).

    Answer

    \(\frac{3}{4}\)

    Example \(\PageIndex{3}\)

    Mulitply: \(\frac{20}{15}·\frac{6}{8}\).

    Answer

    1

    Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, \(x \ne 0\) and \(y \ne 0\).

    Example \(\PageIndex{4}\)

    Mulitply: \(\frac{2x}{3y^2}·\frac{6xy^3}{x^{2}y}\).

    Answer
      .
    Multiply. .
    Factor the numerator and denominator completely, and then remove common factors. .
    Simplify. .

    Example \(\PageIndex{5}\)

    Mulitply: \(\frac{3pq}{q^2}·\frac{5p^{2}q}{6pq}\).

    Answer

    \(\frac{5p^2}{q}\)

    Example \(\PageIndex{6}\)

    Mulitply: \(\frac{6x^{3}y}{7x^2}·\frac{2xy}{3x^{2}y}\).

    Answer

    \(\frac{12y^3}{7}\)

    How to Multiply Rational Expressions

    Example \(\PageIndex{7}\)

    Mulitply: \(\frac{2x}{x^2+x+12}·\frac{x^2−9}{6x^2}\).

    Answer

    The above image has three columns and three rows to show how to multiply rational expressions. Step one is to factor each numerator and denominator completely. Factor x squared minus 9 and x squared plus x plus 12. The rational equation is 2x divided by x squared plus x plus 12 times x squared minus 9 divided by 6x squared, then to 2x divided by x minus 3 times x minus 4 times x minus 3 times x plus 3 divided by 6x squared.Step 2 is to multiply the numerators and denominators. It is helpful to multiply the monomials first. Multiply 2x times x minus 3 times x plus 3 divided by 6x squared times x minus 3 times x minus 4.Step 3 is to divide out the common factors, canceling out 2, x, and x minus 3 in the numerator and 2, x and x minus 3 in the denominator. Leave the denominator in factored form to get x plus 3 divided by 3x times x minus 4.

    Example \(\PageIndex{8}\)

    Mulitply: \(\frac{5x}{x^2+5x+6}·\frac{x^2−4}{10x}\).

    Answer

    \(\frac{x−2}{2(x+3)}\)

    Example \(\PageIndex{9}\)

    Mulitply: \(\frac{9x^2}{x^2+11x+30}·\frac{x^2−36}{3x^2}\).

    Answer

    \(\frac{3(x−6)}{x+5}\)

    Definition: MULTIPLY A RATIONAL EXPRESSION.

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

    Example \(\PageIndex{10}\)

    Multiply: \(\frac{n^2−7n}{n^2+2n+1}·\frac{n+1}{2n}\).

    Answer
      \(\frac{n^2−7n}{n^2+2n+1}·\frac{n+1}{2n}\)
    Factor each numerator and denominator. \(\frac{n(n−7)}{(n+1)(n+1)}·\frac{n+1}{2n}\)
    Multiply the numerators and the denominators. \(\frac{n(n−7)(n+1)}{(n+1)(n+1)2n}\)
    Simplify. \(\frac{n−7}{2(n+1)}\)

    Example \(\PageIndex{11}\)

    Multiply: \(\frac{x^2−25}{x^2−3x−10}·\frac{x+2}{x}\).

    Answer

    \(\frac{x+5}{x}\)

    Example \(\PageIndex{12}\)

    Multiply: \(\frac{x^2−4x}{x^2+5x+6}·\frac{x+2}{x}\).

    Answer

    \(\frac{x−4}{x+3}\)

    Example \(\PageIndex{13}\)

    Multiply: \(\frac{16−4x}{2x−12}·\frac{x^2−5x−6}{x^2−16}\).

    Answer
      \(\frac{16−4x}{2x−12}·\frac{x^2−5x−6}{x^2−16}\)
    Factor each numerator and denominator. \(\frac{4(4−x)}{2(x−6)}·\frac{(x−6)(x+1)}{(x−4)(x+4)}\)
    Multiply the numerators and the denominators. \(\frac{4(4−x)(x−6)(x+1)}{2(x−6)(x−4)(x+4)}\)
    Simplify. \(−\frac{2(x+1)}{(x+4)}\)

    Example \(\PageIndex{14}\)

    Multiply: \(\frac{12x−6x^2}{x^2+8x}·\frac{x^2+11x+24}{x^2−4}\).

    Answer

    \(−\frac{6(x+3)}{x+2}\)

    Example \(\PageIndex{15}\)

    Multiply: \(\frac{9v−3v^2}{9v+36}·\frac{v^2+7v+12}{v^2−9}\).

    Answer

    \(−\frac{v}{3}\)

    Example \(\PageIndex{16}\)

    Multiply: \(\frac{2x−6}{x^2−8x+15}·\frac{x^2−25}{2x+10}\).

    Answer
      .
    Factor each numerator and denominator. .
    Multiply the numerators and denominators. .
    Remove common factors. .
    Simplify. .

    Example \(\PageIndex{17}\)

    Multiply: \(\frac{3a−21}{a^2−9a+14}·\frac{a^2−4}{3a+6}\).

    Answer

    1

    Example \(\PageIndex{18}\)

    Multiply: \(\frac{b^2−b}{b^2+9b−10}·\frac{b^2−100}{b^2−10b}\).

    Answer

    1

    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.

    Remember, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.

    Definition: DIVISION OF RATIONAL EXPRESSIONS

    If p,q,r,s are polynomials where \(q \ne 0\), \(r \ne 0\), \( s \ne 0\)

    \(\frac{p}{q}÷\frac{r}{s}=\frac{p}{q}·\frac{s}{r}\)

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

    How to Divide Rational Expressions

    Example \(\PageIndex{19}\)

    Divide: \(\frac{x+9}{6−x}÷\frac{x^2−81}{x−6}\).

    Answer

    The above image has three columns. It shows the steps to divide rational expressions. Step one is to rewrite the division as the product of the first rational expression and the reciprocal of the second for x plus 9 divided by 6 minus x divided by x squared minus 81 divided by x minus 6. “Flip” the second fraction and change the division sign to multiplication to get x plus 9 divided by 6 minus x times x minus 6 divided by x squared minus 81.Step two is to factor the numerators and denominators completely. Factor x squared minus 81 to get x plus 9 divided by 6 minus x times x minus 6 divided by x minus 9 times x plus 9.Step three is to multiply the numerators and denominators to get x plus 9 times x minus 6 divided by 6 minus x times x minus 9 times x plus 9.Step four is to simplify by dividing out common factors. Divide out the common factors x plus 9, x minus 6 from the numerator and 6 minus x and x plus 9 from the denominator. Remember opposites divide to negative 1. This simplifies to negative 1 divided by x minus 9.

    Example \(\PageIndex{20}\)

    Divide: \(\frac{c+3}{5−c}÷\frac{c^2−9}{c−5}\).

    Answer

    \(−\frac{1}{c−3}\)

    Example \(\PageIndex{21}\)

    Divide: \(\frac{2−d}{d−4}÷\frac{4−d^2}{4−d}\).

    Answer

    \(−\frac{1}{2+d}\)

    Definition: DIVIDE RATIONAL EXPRESSIONS.

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

    Example \(\PageIndex{22}\)

    Divide: \(\frac{3n^2}{n^2−4n}÷\frac{9n^2−45n}{n^2−7n+10}\).

    Answer
      .
    Rewrite the division as the product of the first rational expression and the reciprocal of the second. .
    Factor the numerators and denominators and then multiply. .
    Simplify by dividing out common factors. .
      .

    Example \(\PageIndex{23}\)

    Divide: \(\frac{2m^2}{m^2−8m}÷\frac{8m^2+24m}{m^2+m−6}\).

    Answer

    \(\frac{(m−2)}{4(m−8)}\)

    Example \(\PageIndex{24}\)

    Divide: \(\frac{15n^2}{3n^2+33n}÷\frac{5n−5}{n^2+9n−22}\).

    Answer

    \(\frac{n(n−2)}{n−1}\)

    Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.

    Example \(\PageIndex{25}\)

    Divide: \(\frac{2x^2+5x−12}{x^2−16}÷\frac{2x^2−13x+15}{x^2−8x+16}\).

    Answer
      \(\frac{2x^2+5x−12}{x^2−16}÷\frac{2x^2−13x+15}{x^2−8x+16}\)
    Rewrite the division as the product of the first rational expression and the reciprocal of the second. \(\frac{2x^2+5x−12}{x^2−16}·\frac{x^2−8x+16}{2x^2−13x+15}\)
    Factor the numerators and denominators and then multiply. \(\frac{(2x−3)(x+4)(x−4)(x−4)}{(x−4)(x+4)(2x−3)(x−5)}\)
    Simplify. \(\frac{(x−4)}{(x−5)}\)

    Example \(\PageIndex{26}\)

    Divide: \(\frac{3a^2−8a−3}{a^2−25}÷\frac{3a^2−14a−5}{a^2+10a+25}\).

    Answer

    \(\frac{(a−3)(a+5)}{(a−5)(a−5)}\)

    Exercise \(\PageIndex{27}\)

    Divide: \(\frac{4b^2+7b−2}{1−b^2}÷\frac{4b^2+15b−4}{b^2−2b+1}\).

    Answer

    \(−\frac{(b+2)(b−1)}{(1+b)(b+4)}\)

    Example \(\PageIndex{28}\)

    Divide: \(\frac{p^3+q^3}{2p^2+2pq+2q^2}÷\frac{p^2−q^2}{6}\).

    Answer
      \(\frac{p^3+q^3}{2p^2+2pq+2q^2}÷\frac{p^2−q^2}{6}\)
    Rewrite the division as the product of the first rational expression and the reciprocal of the second. \(\frac{p^3+q^3}{2p^2+2pq+2q^2}·\frac{6}{p^2−q^2}\)
    Factor the numerators and denominators and then multiply. \(\frac{(p+q)(p^2−pq+q^2)6}{2(p^2+pq+q^2)(p−q)(p+q)}\)
    Simplify. \(\frac{3(p^2−pq+q^2)}{(p−q)(p^2+pq+q^2)}\)

    Example \(\PageIndex{29}\)

    Divide: \(\frac{x^3−8}{3x^2−6x+12}÷\frac{x^2−4}{6}\).

    Answer

    \(\frac{2(x^2+2x+4)}{(x+2)(x^2−2x+4)}\)

    Example \(\PageIndex{30}\)

    Divide: \(\frac{2z^2}{z^2−1}÷\frac{z^3−z^2+z}{z^3−1}\).

    Answer

    \(\frac{2z(z^2+z+1)}{(z+1)(z^2−z+1)}\)

    Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide \(\frac{3}{5}÷4\)

    \[\begin{array}{c} {\frac{3}{5}÷4}\\ {\frac{3}{5}÷\frac{4}{1}}\\ {\frac{3}{5}·\frac{1}{4}}\\ \nonumber \end{array}\]

    We do the same thing when we divide rational expressions.

    Example \(\PageIndex{31}\)

    \(\frac{a^2−b^2}{3ab}÷(a^2+2ab+b^2)\).

    Answer
      \(\frac{a^2−b^2}{3ab}÷(a^2+2ab+b^2)\)
    Write the second expression as a fraction. \(\frac{a^2−b^2}{3ab}÷\frac{a^2+2ab+b^2}{1}\)
    Rewrite the division as the first expression times the reciprocal of the second expression. \(\frac{a^2−b^2}{3ab}·\frac{1}{a^2+2ab+b^2}\)
    Factor the numerators and the denominators, and then multiply. \(\frac{(a−b)(a+b)1}{3ab·(a+b)(a+b)}\)
    Simplify. \(\frac{a−b}{3ab(a+b)}\)

    Example \(\PageIndex{32}\)

    \(\frac{2x^2−14x−16}{4}÷(x2+2x+1)\).

    Answer

    \(\frac{x−8}{2(x+1)}\)

    Example \(\PageIndex{33}\)

    \(\frac{y^2−6y+8}{y^2−4y}÷(3y2−12y)\).

    Answer

    \(\frac{y−2}{3y(y−4)}\)

    Example \(\PageIndex{34}\)

    \(\frac{\frac{6x^2−7x+2}{4x−8}}{\frac{2x^2−7x+3}{x^2−5x+6}}\).

    Answer
      \(\frac{\frac{6x^2−7x+2}{4x−8}}{\frac{2x^2−7x+3}{x^2−5x+6}}\)
    Rewrite with a division sign. \(\frac{6x^2−7x+2}{4x−8}÷\frac{2x^2−7x+3}{x^2−5x+6}\)
    Rewrite as product of first times reciprocal of second. \(\frac{6x^2−7x+2}{4x−8}·\frac{x^2−5x+6}{2x^2−7x+3}\)
    Factor the numerators and the denominators, and then multiply \(\frac{(2x−1)(3x−2)(x−2)(x−3)}{4(x−2)(2x−1)(x−3)}\)
    Simplify. \(\frac{3x−2}{4}\)

    Example \(\PageIndex{35}\)

    \(\frac{\frac{3x^2+7x+2}{4x+24}}{\frac{3x^2−14x−5}{x^2+x−30}}\).

    Answer

    \(\frac{x+2}{4}\)

    Example \(\PageIndex{36}\)

    \(\frac{\frac{y^2−36}{2y^2+11y−6}}{\frac{2y^2−2y−60}{8y−4}}\).

    Answer

    \(\frac{2}{y+5}\)

    ​​​​​​​If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.

    Example \(\PageIndex{37}\)

    \(\frac{3x−6}{4x−4}·\frac{x^2+2x−3}{x^2−3x−10}÷\frac{2x+12}{8x+16}\).

    Answer
      .
    Rewrite the division as multiplication by the reciprocal. .
    Factor the numerators and the denominators, and then multiply. .
    Simplify by dividing out common factors. .
    Simplify. .

    Example \(\PageIndex{38}\)

    \(\frac{4m+4}{3m−15}·\frac{m^2−3m−10}{m^2−4m−32}÷\frac{12m−36}{6m−48}\).

    Answer

    \(\frac{2(m+1)(m+2)}{3(m+4)(m−3)}\)​​​​​​​

    Example \(\PageIndex{39}\)

    \(\frac{2n^2+10n}{n−1}÷\frac{n^2+10n+24}{n^2+8n−9}·\frac{n+4}{8n^2+12n}\).

    Answer

    \(\frac{(n+5)(n+9)}{2(n+6)(2n+3)}\)

    Key Concepts

    • Multiplication of Rational Expressions
      • If p,q,r,s are polynomials where \(q \ne 0\) and \(s \ne 0\), then \(\frac{p}{q}·\frac{r}{s}=\frac{pr}{qs}\)

      • To multiply rational expressions, multiply the numerators and multiply the denominators
    • Multiply a Rational Expression
      1. Factor each numerator and denominator completely.
      2. Multiply the numerators and denominators.
      3. Simplify by dividing out common factors.
    • Division of Rational Expressions
      • If p,q,r,s are polynomials where \(q \ne 0\), \(r \ne 0\), \( s \ne 0\), then \(\frac{p}{q}÷\frac{r}{s}=\frac{p}{q}·\frac{s}{r}\)

      • To divide rational expressions multiply the first fraction by the reciprocal of the second.
    • Divide Rational Expressions
      1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
      2. Factor the numerators and denominators completely.
      3. Multiply the numerators and denominators together.
      4. Simplify by dividing out common factors.