10.4: Multiply Polynomials (Part 1)

Last updated
Save as PDF

Page ID: 21770

$ \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.

Distribute: 2(x + 3). If you missed the problem, review Example 7.4.1.
Distribute: −11(4 − 3a). If you missed the problem, review Example 7.4.10.
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

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."

Search

Learning Objectives

be prepared!

Multiply a Polynomial by a Monomial

Example \(\PageIndex{1}\):

Exercise \(\PageIndex{1}\):

Exercise \(\PageIndex{2}\):

Example \(\PageIndex{2}\):

Exercise \(\PageIndex{3}\):

Exercise \(\PageIndex{4}\):

Example \(\PageIndex{3}\):

Exercise \(\PageIndex{5}\):

Exercise \(\PageIndex{6}\):

Example \(\PageIndex{4}\):

Exercise \(\PageIndex{7}\):

Exercise \(\PageIndex{8}\):

Example \(\PageIndex{5}\):

Exercise \(\PageIndex{9}\):

Exercise \(\PageIndex{10}\):

Example \(\PageIndex{6}\):

Exercise \(\PageIndex{11}\):

Exercise \(\PageIndex{12}\):

Multiply a Binomial by a Binomial

Using the Distributive Property

Example \(\PageIndex{7}\):

Exercise \(\PageIndex{13}\):

Exercise \(\PageIndex{14}\):

Example \(\PageIndex{8}\):

Exercise \(\PageIndex{15}\):

Exercise \(\PageIndex{16}\):

Example \(\PageIndex{9}\):

Exercise \(\PageIndex{17}\):

Exercise \(\PageIndex{18}\):

Example \(\PageIndex{10}\):

Exercise \(\PageIndex{19}\):

Exercise \(\PageIndex{20}\):

Contributors and Attributions