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1.12: Dividing Polynomials

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    In the previous chapter, we added, subtracted, and multiplied polynomials. Now, what remains is dividing polynomials. We will only consider division of a polynomial by a monomial. The division of a monomial by a monomial was already considered in Chapter 5, which we now recall.

    Rule \(\PageIndex{1}\)

    We recall the rules for dividing variables.

    \[\dfrac{x^{n}}{x^{m}}=x^{n-m}\]

    for any integers \(n\) and \(m\)

    Example \(\PageIndex{2}\)

    Simplify.

    1. \(\dfrac{27 x^{3} y^{5}}{3 x^{2} y^{3}}=\dfrac{27 x^{3-2} y^{5-3}}{3}=9 x^{1} y^{2}=9 x y^{2}\)
    2. \(\dfrac{-56 a^{8} b^{6} c^{4}}{-7 a^{5} b c^{4}}=\dfrac{-56 a^{8-5} b^{6-1} c^{4-4}}{-7}=8 a^{3} b^{5} c^{0}=8 a^{3} b^{5}\)

    Of course, when the power of a variable is higher in the denominator than in the numerator, then those variables will remain in the denominator, just as we did in chapter 5.

    Example \(\PageIndex{3}\)

    Simplify.

    1. \(\dfrac{42 p^{7} q^{4}}{-3 p^{3} q^{2}}=\dfrac{42}{-3} q^{4-2} p^{7-4}=\dfrac{14}{-1} q^{2} p^{3}=-14 q^{2} p^{3}\)
    2. \(\dfrac{24 r^{4} s^{9} t^{5}}{20 r s^{6} t^{2}}=\dfrac{24}{20} r^{4-1} s^{9-6} t^{5-2}=\dfrac{6}{5} r^{3} s^{3} t^{3}\)

    We now study how a polynomial can be divided by a monomial. Recall the usual rule for adding fractions with common denominator.

    Same Denominator Fractions

    Fractions with common denominator can be added (or subtracted) by adding (or subtracting) the numerators:

    • Add: \[\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c}\]
    • Subtract: \[\dfrac{a}{c}-\dfrac{b}{c}=\dfrac{a-b}{c}\]

    Reversing the above rule helps us to divide a polynomial by a monomial.

    Example \(\PageIndex{4}\)

    Simplify as much as possible.

    a) \(\dfrac{6 x+15}{3}=\dfrac{6 x}{3}+\dfrac{15}{3}=2 x+5\)

    b) \(\dfrac{14 x^{3}-8 x^{2}}{2 x}=\dfrac{14 x^{3}}{2 x}-\dfrac{8 x^{2}}{2 x}=7 x^{2}-4 x\)

    c) \(\begin{align*}
    \dfrac{14 y^{6}-28 y^{5}+21 y^{3}}{-7 y^{2}} &=\dfrac{14 y^{6}}{-7 y^{2}}-\dfrac{28 y^{5}}{-7 y^{2}}+\dfrac{21 y^{3}}{-7 y^{2}} \\
    &=-2 y^{4}-\left(-4 y^{3}\right)-3 y \\
    &=-2 y^{4}+4 y^{3}-3 y
    \end{align*}\)

    d) \(\begin{align*}
    \dfrac{a^{2} b^{4}-4 a b^{3}-2 a^{4} b^{2}}{a b^{2}} &=\dfrac{a^{2} b^{4}}{a b^{2}}-\dfrac{4 a b^{3}}{a b^{2}}-\dfrac{2 a^{4} b^{2}}{a b^{2}} \\
    &=a b^{2}-4 b-2 a^{3}
    \end{align*}\)

    e) \(\begin{align*}
    \dfrac{-6 r^{5} t^{4}+30 r^{4} s^{2} t^{5}-42 r^{3} s^{2} t^{3}}{-6 r t^{3}} &=\dfrac{-6 r^{5} t^{4}}{-6 r t^{3}}+\dfrac{30 r^{4} s^{2} t^{5}}{-6 r t^{3}}-\dfrac{42 r^{3} s^{2} t^{3}}{-6 r t^{3}} \\
    &=r^{4} t-5 r^{3} s^{2} t^{2}-\left(-7 r^{2} s^{2}\right) \\
    &=r^{4} t-5 r^{3} s^{2} t^{2}+7 r^{2} s^{2}
    \end{align*}\)

    Exit Problem

    Simplify: \(\dfrac{27 x^{2} y-3 x y+15 x y^{2}}{-3 x y}\)

    This page titled 1.12: Dividing Polynomials is shared under a CC BY-NC-ND 4.0 license and was authored, remixed, and/or curated by Samar ElHitti, Marianna Bonanome, Holly Carley, Thomas Tradler, & Lin Zhou (New York City College of Technology at CUNY Academic Works) .