Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

6.4: Multiply polynomial expressions

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

We can multiply polynomials of different forms, but the method will be the same. We first look at multiplying monomials, multiplying a monomial and polynomial, and then finish with multiplying polynomials. We are using the product rule of exponents and the distributive property.

Multiply a Polynomial Expression by a Monomial

Example 6.4.1

Multiply: (4x3y4z)(2x2y6z3)

Solution

(4x3y4z)(2x2y6z3)Rewrite without parenthesis4x3y4z2x2y6z3Multiply coefficients and apply the product rule of exponents42x3+2y4+6z1+3Simplify8x5y10z4Product

Example 6.4.2

Multiply: 4x3(5x22x+5)

Solution

4x3(5x22x+5)Distribute 4x34x35x24x32x+4x35Multiply and apply the product rule of exponents20x58x4+20x3Product

Example 6.4.3

Multiply: 2a3b(3ab24a)

Solution

2a3b(3ab24a)Distribute 2a3b2a3b3ab22a3b4aMultiply and apply the product rule of exponents6a3+1b1+28a3+1bSimplify6a4b38a4bProduct

Multiplying with Binomials

There are several different methods for multiplying polynomials, all of which result in the same answer. We discuss multiplying by distribution and the FOIL method.

Example 6.4.4

Multiply: (3x+5)(x+13)

Solution

We will multiply using distribution and then simplify.

(3x+5)(x+13)Distribute 3x and 5 to (x+13)3x(x+13)+5(x+13)Distribute3x2+39x+5x+65Combine like terms3x2+44x+65Product

Example 6.4.5

Multiply: (4x+7y)(3x2y)

Solution

We will multiply using the FOIL method. FOIL is an acronym and represents

First-Multiply the first terms in each parenthesisOuter-Multiply the outer terms in each parenthesisInner-Multiply the inner terms in each parenthesisLast-Multiply the last terms in each parenthesis

clipboard_eaeb1e46019594ad0d1a1ee737c0f5588.png

=12x28xy+21xy14y2=12x2+13xy+14y2

Multiplying with Trinomials

A trinomial is a polynomial with three terms. Usually, in Algebra, a trinomial takes the form of ax2+bx+c, where a,b, and c are coefficients.

Example 6.4.6

Multiply: (2x5)(4x27x+3)

Solution

Since we are multiplying a binomial with a trinomial, we can use distribution to multiply.

(2x5)(4x27x+3)Distribute 2x and 5 to (4x27x+3)2x(4x27x+3)5(4x27x+3)Distribute8x314x2+6x20x2+35x15Combine like terms8x334x2+41x15Product

Example 6.4.7

Multiply: (5x2+x10)(3x210x6)

Solution

Since we are multiplying a trinomial with a trinomial, then we can use distribution to multiply.

(5x2+x10)(3x210x6)Distribute 5x2,x, and 10to (3x210x6)5x2(3x210x6)+x(3x210x6)10(3x210x6)Distribute15x450x330x2+3x310x26x30x2+100x+60Combine like terms15x447x370x2+94x+60Product

Multiplying Monomials and Binomials

Example 6.4.8

Multiply: 3x(2x4)(x+5)

Solution

We first use FOIL to multiply the binomials and then distribute the 3x.

clipboard_e46c3960200bc28bb16a3fc94d2dd98c6.png

=3x(2x2+10x4x20)=3x(2x2+6x20)

Lastly, we distribute 3x:

3x(2x2+6x20)3x2x2+3x6x3x206x3+18x260x

Thus, the product is 6x3+18x260x.

Note

In Example 6.4.8 , a common error is distributing the 3x first and into both parenthesis. While we can distribute the 3x into the (2x4) factor, we cannot distribute into both factors. We recommend to multiply the binomials first, then distribute the monomial last.

Multiplying Polynomial Functions

We can multiply two polynomial functions the same way we multiply polynomial expressions, except, now, we have functions. The method is the same, but the notation and problems change.

Definition

If f and g are two functions of x, then (fg)(x)=f(x)g(x)

where x is in the domain of f and g.

Example 6.4.9

Let f(x)=2x1 and g(x)=x+4. Find (fg)(x).

Solution

We start by applying the definition, then simplify completely.

(fg)(x)=f(x)g(x)Apply the definition(fg)(x)=(2x1)(x+4)Multiply two binomials(fg)(x)=2x2+8xx4Combine like terms(fg)(x)=2x2+7x4The product of f and g

Multiply Polynomial Expressions Homework

Multiply and simplify.

Exercise 6.4.1

6(p7)

Exercise 6.4.2

2(6x+3)

Exercise 6.4.3

5m4(4m+4)

Exercise 6.4.4

(4n+6)(8n+8)

Exercise 6.4.5

(8b+3)(7b5)

Exercise 6.4.6

(4x+5)(2x+3)

Exercise 6.4.7

(3v4)(5v2)

Exercise 6.4.8

(6x7)(4x+1)

Exercise 6.4.9

(5x+y)(6x4y)

Exercise 6.4.10

(x+3y)(3x+4y)

Exercise 6.4.11

(7x+5y)(8x+3y)

Exercise 6.4.12

(r7)(6r2r+5)

Exercise 6.4.13

(6n4)(2n22n+5)

Exercise 6.4.14

(6x+3y)(6x27xy+4y2)

Exercise 6.4.15

(8n2+4n+6)(6n25n+6)

Exercise 6.4.16

(5k2+3k+3)(3k2+3k+6)

Exercise 6.4.17

3(3x4)(2x+1)

Exercise 6.4.18

3(2x+1)(4x5)

Exercise 6.4.19

7(x5)(x2)

Exercise 6.4.20

6(4x1)(4x+1)

Exercise 6.4.21

4k(8k+4)

Exercise 6.4.22

3n2(6n+7)

Exercise 6.4.23

3(4r7)

Exercise 6.4.24

(2x+1)(x4)

Exercise 6.4.25

(r+8)(4r+8)

Exercise 6.4.26

(7n6)(n+7)

Exercise 6.4.27

(6a+4)(a8)

Exercise 6.4.28

(5x6)(4x1)

Exercise 6.4.29

(2u+3v)(8u7v)

Exercise 6.4.30

(8u+6v)(5u8v)

Exercise 6.4.31

(5a+8b)(a3b)

Exercise 6.4.32

(4x+8)(4x2+3x+5)

Exercise 6.4.33

(2b3)(4b2+4b+4)

Exercise 6.4.34

(3m2n)(7m2+6mn+4n2)

Exercise 6.4.35

(2a2+6a+3)(7a26a+1)

Exercise 6.4.36

(7u2+8uv6v2)(6u2+4uv+3v2)

Exercise 6.4.37

5(x4)(2x3)

Exercise 6.4.38

2x(4x+1)(2x6)

Exercise 6.4.39

5x(2x1)(4x+1)

Exercise 6.4.40

3x2(2x+3)(6x+9)

Perform the indicated operations given the set of functions.

Exercise 6.4.41

Let g(x)=4x+5 and h(x)=x2+5x, find (gh)(x).

Exercise 6.4.42

Let p(t)=t4 and r(t)=2t, find (pr)(t).

Exercise 6.4.43

Let f(n)=2n25n and k(n)=n+5, find (fk)(n).


This page titled 6.4: Multiply polynomial expressions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?