Skip to main content
Mathematics LibreTexts

10.4: Multiply Polynomials (Part 1)

  • Page ID
    7253
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Learning Objectives
    • Multiply a polynomial by a monomial
    • Multiply a binomial by a binomial
    • Multiply a trinomial by a binomial
    be prepared!

    Before you get started, take this readiness quiz.

    1. Distribute: 2(x + 3). If you missed the problem, review Example 7.4.1.
    2. Distribute: −11(4 − 3a). If you missed the problem, review Example 7.4.10.
    3. Combine like terms: x2 + 9x + 7x + 63. If you missed the problem, review Example 2.3.9.

    Multiply a Polynomial by a Monomial

    In Distributive Property you learned to use the Distributive Property to simplify expressions such as 2(x − 3). You multiplied both terms in the parentheses, x and 3, by 2, to get 2x − 6. With this chapter's new vocabulary, you can say you were multiplying a binomial, x − 3, by a monomial, 2. Multiplying a binomial by a monomial is nothing new for you!

    Example \(\PageIndex{1}\):

    Multiply: 3(x + 7).

    Solution

    Distribute. CNX_BMath_Figure_10_03_001_img-01.png
      3 • x + 3 • 7
    Simplify. 3x + 21
    Exercise \(\PageIndex{1}\):

    Multiply: 6(x + 8).

    Answer

    6x + 48

    Exercise \(\PageIndex{2}\):

    Multiply: 2(y + 12).

    Answer

    2y + 24

    Example \(\PageIndex{2}\):

    Multiply: x(x − 8).

    Solution

    Distribute. CNX_BMath_Figure_10_03_044_img-01.png
      x2 - 8x
    Simplify. x2 - 8x
    Exercise \(\PageIndex{3}\):

    Multiply: y(y − 9).

    Answer

    \( y^{2}-9 y\)

    Exercise \(\PageIndex{4}\):

    Multiply: p(p − 13).

    Answer

    \( p^2 - 13p\)

    Example \(\PageIndex{3}\):

    Multiply: 10x(4x + y).

    Solution

    Distribute. CNX_BMath_Figure_10_03_045_img-02.png
      10x • 4x + 10x • y
    Simplify. 40x2 + 10xy
    Exercise \(\PageIndex{5}\):

    Multiply: 8x(x + 3y).

    Answer

    \(8x^2+24xy \)

    Exercise \(\PageIndex{6}\):

    Multiply: 3r(6r + s).

    Answer

    \(18r^2+3rs \)

    Multiplying a monomial by a trinomial works in much the same way.

    Example \(\PageIndex{4}\):

    Multiply: −2x(5x2 + 7x − 3).

    Solution

    Distribute. CNX_BMath_Figure_10_03_046_img-01.png
      -2x • 5x2 + (-2x) • 7x - (-2x) • 3
    Simplify. -10x3 -14x2 + 6x
    Exercise \(\PageIndex{7}\):

    Multiply: −4y(8y2 + 5y − 9).

    Answer

    \(-32y^3-20y^2+36y \)

    Exercise \(\PageIndex{8}\):

    Multiply: −6x(9x2 + x − 1).

    Answer

    \( -54x^3-6x^2+6x\)

    Example \(\PageIndex{5}\):

    Multiply: 4y3(y2 − 8y + 1).

    Solution

    Distribute. CNX_BMath_Figure_10_03_047_img-01.png
      4y3 • y2 - 4y3 • 8y + 4y3 • 1
    Simplify. 4y5 -32y4 + 4y3
    Exercise \(\PageIndex{9}\):

    Multiply: 3x2 (4x2 − 3x + 9).

    Answer

    \( 12 x^{4}-9 x^{3}+27 x^{2}\)

    Exercise \(\PageIndex{10}\):

    Multiply: 8y2 (3y2 − 2y − 4).

    Answer

    \(24 y^{4}-16 y^{3}-32 y^{2} \)

    Now we will have the monomial as the second factor.

    Example \(\PageIndex{6}\):

    Multiply: (x + 3)p.

    Solution

    Distribute. CNX_BMath_Figure_10_03_048_img-01.png
      x • p + 3 • p
    Simplify. xp + 3p
    Exercise \(\PageIndex{11}\):

    Multiply: (x + 8)p.

    Answer

    \(xp+8p \)

    Exercise \(\PageIndex{12}\):

    Multiply: (a + 4)p.

    Answer

    \(ap + 4p \)

    Multiply a Binomial by a Binomial

    Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.

    Using the Distributive Property

    We will start by using the Distributive Property. Look again at Example \(\PageIndex{6}\).

      CNX_BMath_Figure_10_03_049_img-01.png
    We distributed the p to get CNX_BMath_Figure_10_03_049_img-02.png
    What if we have (x + 7) instead of p? Think of the (x + 7) as the \(\textcolor{red}{p}\) above. CNX_BMath_Figure_10_03_049_img-03.png
    Distribute (x + 7). CNX_BMath_Figure_10_03_049_img-04.png
    Distribute again. x2 + 7x + 3x + 21
    Combine like terms. x2 + 10x + 21

    Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

    Be careful to distinguish between a sum and a product.

    \[\begin{split} &\textbf{Sum} \qquad \qquad \qquad \quad \textbf{Product} \\ &x + x \qquad \qquad \qquad \qquad x \cdot x \\ &\; \; 2x \qquad \qquad \qquad \qquad \qquad x^{2} \\ combine\; &like\; terms \qquad add\; exponents\; of\; like\; bases \end{split}\]

    Example \(\PageIndex{7}\):

    Multiply: (x + 6)(x + 8).

    Solution

      CNX_BMath_Figure_10_03_050_img-01.png
    Distribute (x + 8). CNX_BMath_Figure_10_03_050_img-02.png
    Distribute again. x2 + 8x + 6x + 48
    Simplify. x2 + 14x + 48
    Exercise \(\PageIndex{13}\):

    Multiply: (x + 8)(x + 9).

    Answer

    \(x^{2}+17 x+72 \)

    Exercise \(\PageIndex{14}\):

    Multiply: (a + 4)(a + 5).

    Answer

    \(a^{2}+9 a+20 \)

    Now we'll see how to multiply binomials where the variable has a coefficient.

    Example \(\PageIndex{8}\):

    Multiply: (2x + 9)(3x + 4).

    Solution

    Distribute (3x + 4). CNX_BMath_Figure_10_03_051_img-01.png
    Distribute again. 6x2 + 8x + 27x + 36
    Simplify. 6x2 + 35x + 36
    Exercise \(\PageIndex{15}\):

    Multiply: (5x + 9)(4x + 3).

    Answer

    \(20 x^{2}+51 x+27 \)

    Exercise \(\PageIndex{16}\):

    Multiply: (10m + 9)(8m + 7).

    Answer

    \(80 m^{2}+142 m+63 \)

    In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.

    Example \(\PageIndex{9}\):

    Multiply: (4y + 3)(6y − 5).

    Solution

    Distribute. CNX_BMath_Figure_10_03_052_img-01.png
    Distribute again. 24y2 − 20y + 18y − 15
    Simplify. 24y2 − 2y − 15
    Exercise \(\PageIndex{17}\):

    Multiply: (7y + 1)(8y − 3).

    Answer

    \( 56 y^{2}-13 y-3\)

    Exercise \(\PageIndex{18}\):

    Multiply: (3x + 2)(5x − 8).

    Answer

    \(15 x^{2}-14 x-16 \)

    Up to this point, the product of two binomials has been a trinomial. This is not always the case.

    Example \(\PageIndex{10}\):

    Multiply: (x + 2)(x − y).

    Solution

    Distribute. CNX_BMath_Figure_10_03_053_img-02.png
    Distribute again. x2 - xy + 2x - 2y
    Simplify. There are no like terms to combine.
    Exercise \(\PageIndex{19}\):

    Multiply: (x + 5)(x − y).

    Answer

    \( x^{2}-x y+5 x-5 y\)

    Exercise \(\PageIndex{20}\):

    Multiply: (x + 2y)(x − 1).

    Answer

    \( x^{2}-x+2 x y-2 y\)

    Contributors and Attributions


    This page titled 10.4: Multiply Polynomials (Part 1) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

    • Was this article helpful?