1.12: Dividing Polynomials
- Page ID
- 45250
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
We recall the rules for dividing variables.
\[\dfrac{x^{n}}{x^{m}}=x^{n-m}\]
for any integers \(n\) and \(m\)
Simplify.
- \(\dfrac{27 x^{3} y^{5}}{3 x^{2} y^{3}}=\dfrac{27 x^{3-2} y^{5-3}}{3}=9 x^{1} y^{2}=9 x y^{2}\)
- \(\dfrac{-56 a^{8} b^{6} c^{4}}{-7 a^{5} b c^{4}}=\dfrac{-56 a^{8-5} b^{6-1} c^{4-4}}{-7}=8 a^{3} b^{5} c^{0}=8 a^{3} b^{5}\)
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.
Simplify.
- \(\dfrac{42 p^{7} q^{4}}{-3 p^{3} q^{2}}=\dfrac{42}{-3} q^{4-2} p^{7-4}=\dfrac{14}{-1} q^{2} p^{3}=-14 q^{2} p^{3}\)
- \(\dfrac{24 r^{4} s^{9} t^{5}}{20 r s^{6} t^{2}}=\dfrac{24}{20} r^{4-1} s^{9-6} t^{5-2}=\dfrac{6}{5} r^{3} s^{3} t^{3}\)
We now study how a polynomial can be divided by a monomial. Recall the usual rule for adding fractions with common denominator.
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.
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) \(\begin{align*}
\dfrac{-6 r^{5} t^{4}+30 r^{4} s^{2} t^{5}-42 r^{3} s^{2} t^{3}}{-6 r t^{3}} &=\dfrac{-6 r^{5} t^{4}}{-6 r t^{3}}+\dfrac{30 r^{4} s^{2} t^{5}}{-6 r t^{3}}-\dfrac{42 r^{3} s^{2} t^{3}}{-6 r t^{3}} \\
&=r^{4} t-5 r^{3} s^{2} t^{2}-\left(-7 r^{2} s^{2}\right) \\
&=r^{4} t-5 r^{3} s^{2} t^{2}+7 r^{2} s^{2}
\end{align*}\)
Simplify: \(\dfrac{27 x^{2} y-3 x y+15 x y^{2}}{-3 x y}\)