Skip to main content
Mathematics LibreTexts

8: Dividing Polynomials

  • Page ID
    49002
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    We now start our discussion of specific classes of examples of functions. The first class of functions that we discuss are polynomials and rational functions. Let us first recall the definition of polynomials and rational functions.

    Definition: Monomial and Polynomial

    A monomial is a number, a variable, or a product of numbers and variables. A polynomial is a sum (or difference) of monomials.

    Example \(\PageIndex{1}\)

    Examples of monomials and polynomials.

    Solution

    The following are examples of monomials:

    \[5,\quad x,\quad 7x^2 y,\quad -12 x^3y^2 z^4, \quad\sqrt{2}\cdot a^3 n^2 x y \nonumber \]

    The following are examples of polynomials:

    \[x^2+3x-7,\quad 4x^2y^3+2x+z^3+4mn^2,\quad -5x^3-x^2-4x-9, \quad 5x^2y^4 \nonumber \]

    In particular, every monomial is also a polynomial.

    We are mainly interested in polynomials in one variable \(x\), and consider these as functions. For example, \(f(x)=x^2+3x-7\) is such a function.

    Definition: Polynomial Function

    A polynomial is a function \(f\) of the form

    \[f(x)=a_n x^n+a_{n-1} x^{n-1}+\dots +a_2 x^2+ a_1 x + a_0 \nonumber \]

    for some real (or complex) numbers \(a_0, a_1,\dots, a_n\). The domain of a polynomial \(f\) is all real numbers (see our standard convention definition).

    The numbers \(a_0, a_1,\dots, a_n\) are called coefficients. For each \(k\), the number \(a_k\) is the coefficient of \(x^k\). The number \(a_n\) is called the leading coefficient and \(n\) is the degree of the polynomial.

    The zeros of a polynomial are usually referred to as roots. Therefore \(x\) is a root of a polynomial \(f\) precisely when \(f(x)=0\).

    Definition: Rational Function

    A rational function is a fraction of two polynomials \(f(x)=\dfrac{g(x)}{h(x)}\), where \(g(x)\) and \(h(x)\) are both polynomials. The domain of \(f\) is all real numbers for which the denominator \(h(x)\) is not zero:

    \[D_f\,\,=\,\,\{\,\, x \,\,| \,\, h(x)\neq 0 \,\, \} \nonumber \]

    The following are examples of rational functions:

    \[f(x)=\dfrac{-3x^2+7x-5}{2x^3+4x^2+3x+1}, \quad f(x)=\dfrac{1}{x}, \quad f(x)=-x^2+3x+5 \nonumber \]


    This page titled 8: Dividing Polynomials is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

    • Was this article helpful?