10.3 Multiply polynomials

Prealgebra Page 1 / 3

By the end of this section, you will be able to:

Multiply a polynomial by a monomial
Multiply a binomial by a binomial
Multiply a trinomial by a binomial

Before you get started, take this readiness quiz.

Distribute: $2 (x + 3) .$
If you missed the problem, review Distributive Property .
Distribute: $−11 (4 - 3 a) .$
If you missed the problem, review Distributive Property .
Combine like terms: $x^{2} + 9 x + 7 x + 63 .$
If you missed the problem, review Evaluate, Simplify and Translate Expressions .

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 $2 x - 6 .$ 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!

Multiply: $3 (x + 7) .$

Solution

	$3 (x + 7)$
Distribute.
	$3 \cdot x + 3 \cdot 7$
Simplify.	$3 x + 21$

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Multiply: $6 (x + 8) .$

6 x + 48

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Multiply: $2 (y + 12) .$

2 y + 24

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Multiply: $x (x - 8) .$

Solution


Distribute.

Simplify.

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Multiply: $y (y - 9) .$

y ² − 9 y

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Multiply: $p (p - 13) .$

p ² − 13 p

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Multiply: $10 x (4 x + y) .$

Solution


Distribute.

Simplify.

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Multiply: $8 x (x + 3 y) .$

8 x ² + 24 xy

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Multiply: $3 r (6 r + s) .$

18 r ² + 3 rs

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

Multiply: $−2 x (5 x^{2} + 7 x - 3) .$

Solution

	$−2 x (5 x^{2} + 7 x - 3)$
Distribute.
	$−2 x \cdot 5 x^{2} + (−2 x) \cdot 7 x - (−2 x) \cdot 3$
Simplify.	$−10 x^{3} - 14 x^{2} + 6 x$

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Multiply: $−4 y (8 y^{2} + 5 y - 9) .$

−32 y ³ − 20 y ² + 36 y

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Multiply: $−6 x (9 x^{2} + x - 1) .$

−54 x ³ − 6 x ² + 6 x

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Multiply: $4 y^{3} (y^{2} - 8 y + 1) .$

Solution

	$4 y^{3} (y^{2} - 8 y + 1)$
Distribute.
	$4 y^{3} \cdot y^{2} - 4 y^{3} \cdot 8 y + 4 y^{3} \cdot 1$
Simplify.	$4 y^{5} - 32 y^{4} + 4 y^{3}$

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Multiply: $3 x^{2} (4 x^{2} - 3 x + 9) .$

12 x ⁴ − 9 x ³ + 27 x ²

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Multiply: $8 y^{2} (3 y^{2} - 2 y - 4) .$

24 y ⁴ − 16 y ³ − 32 y ²

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Now we will have the monomial as the second factor.

Multiply: $(x + 3) p .$

Solution

	$(x + 3) p$
Distribute.
	$x \cdot p + 3 \cdot p$
Simplify.	$x p + 3 p$

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Multiply: $(x + 8) p .$

xp + 8 p

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Multiply: $(a + 4) p .$

ap + 4 p

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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 [link] .


We distributed the $p$ to get
What if we have $(x + 7)$ instead of $p$ ?
Distribute $(x + 7)$ .
Distribute again.	$x^{2} + 7 x + 3 x + 21$
Combine like terms.	$x^{2} + 10 x + 21$

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{matrix} Sum & Product \\ x + x & x \cdot x \\ 2 x & x^{2} \\ combine like terms & add exponents of like bases \end{matrix}

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

Solution

	$(x + 6) (x + 8)$

Distribute $(x + 8)$ .
Distribute again.	$x^{2} + 8 x + 6 x + 48$
Simplify.	$x^{2} + 14 x + 48$

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

x ² + 17 x + 72

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

a ² + 9 a + 20

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

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

Solution

	$(2 x + 9) (3 x + 4)$
Distribute. $(3 x + 4)$ .
Distribute again.	$6 x^{2} + 8 x + 35 x + 36$
Simplify.	$6 x^{2} + 35 x + 36$

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

20 x ² + 51 x + 27

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

80 m ² + 142 m + 63

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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.

Multiply: $(4 y + 3) (6 y - 5) .$

Solution

	$(4 y + 3) (6 y - 5)$
Distribute.
Distribute again.	$24 y^{2} - 20 y + 18 y - 15$
Simplify.	$24 y^{2} - 2 y - 15$

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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