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Multiplying Polynomials

Author: Sophia
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what's covered
In this lesson, you will learn how to multiply three binomials. Specifically, this lesson will cover:

Table of Contents

1. Distribution

When multiplying two expressions, we can distribute all factors into each term of the other expression. With simple algebraic expression, this typically involves a coefficient and a single variable, as in the example below:

EXAMPLE

Distribute short dash 4 open parentheses x plus 6 close parentheses.

short dash 4 open parentheses x plus 6 close parentheses Distribute -4 into each term in the parentheses
short dash 4 times x plus open parentheses short dash 4 close parentheses times 6 Evaluate the multiplication
short dash 4 x minus 24 Our solution

We can also use the distribution property with examples involving other variables or exponents:

EXAMPLE

Multiply 5 x open parentheses 3 y plus x squared close parentheses.

5 x open parentheses 3 y plus x squared close parentheses Distribute 5x into each term in the parentheses
5 x times 3 y plus 5 x times x squared Evaluate the multiplication
15 x y plus 5 x cubed Rewrite in descending order of degree
5 x cubed plus 15 x y Our solution

2. FOIL

In the previous examples, we multiplied a monomial (single-term expression) by a binomial (two-term expression). When multiplying two binomials together, we use a special case of the distributive rule, commonly referred to as FOIL. As we have learned earlier in the course, FOIL stands for First, Outside, Inside, Last, and describes how to distribute all terms in binomial multiplication.

EXAMPLE

Multiply open parentheses 2 x minus 3 close parentheses open parentheses x plus 4 close parentheses.

open parentheses 2 x minus 3 close parentheses open parentheses x plus 4 close parentheses Multiply the first terms to start the calculation, 2 x times x equals 2 x squared
2 x squared Multiply the outside terms, 2 x times 4 equals 8, and add to the calculation
2 x squared plus 8 x Multiply the inside terms, short dash 3 times x equals short dash 3 x, and add to the calculation
2 x squared plus 8 x minus 3 x Multiply the last terms, short dash 3 times 4 equals short dash 12, and add to the calculation
2 x squared plus 8 x minus 3 x minus 12 Combine like terms, 8x and -3x
2 x squared plus 5 x minus 12 Our solution

hint
As you are multiplying with FOIL, you can also just write this as one, long expression and then evaluate:

table attributes columnalign left end attributes row cell open parentheses 2 x minus 3 close parentheses open parentheses x plus 4 close parentheses end cell row cell stack open parentheses 2 x close parentheses open parentheses x close parentheses with f i r s t below plus stack open parentheses 2 x close parentheses open parentheses 4 close parentheses with o u t s i d e below plus stack open parentheses short dash 3 close parentheses open parentheses x close parentheses with i n s i d e below plus stack open parentheses short dash 3 close parentheses open parentheses 4 close parentheses with l a s t below end cell row cell 2 x squared plus 8 x minus 3 x minus 12 end cell row cell 2 x squared minus 5 x minus 12 end cell end table

3. Multiplying Three Binomials

How can the distributive and FOIL processes be modified to multiply three binomials? One strategy is to perform the steps in FOIL to two of the binomials and then distribute the third binomial into the product created by FOIL.

EXAMPLE

Multiply open parentheses 2 x minus 1 close parentheses open parentheses x plus 1 close parentheses open parentheses x minus 3 close parentheses.

open parentheses 2 x minus 1 close parentheses open parentheses x plus 1 close parentheses open parentheses x minus 3 close parentheses Choose two binomials to FOIL, for instance, open parentheses x plus 1 close parentheses and open parentheses x minus 3 close parentheses
open parentheses 2 x minus 1 close parentheses open parentheses x plus 1 close parentheses open parentheses x minus 3 close parentheses FOIL the two binomials
open parentheses 2 x minus 1 close parentheses open parentheses stack open parentheses x times x close parentheses with f i r s t below plus stack open parentheses x times short dash 3 close parentheses with o u t s i d e below plus stack open parentheses 1 times x close parentheses with i n s i d e below plus stack open parentheses 1 times short dash 3 close parentheses with l a s t below close parentheses Evaluate the multiplication
open parentheses 2 x minus 1 close parentheses open parentheses x squared minus 3 x plus x minus 3 close parentheses Combine like terms
open parentheses 2 x minus 1 close parentheses open parentheses x squared minus 2 x minus 3 close parentheses The solution to the two binomials

So far, all that we have done is used FOIL to multiply two of the three binomials. In order to multiply the third binomial, we will distribute each term of 2 x minus 1 into every term in x squared minus 2 x minus 3. In order to keep things organized, it is helpful to distribute separately, and then add the two new polynomials:

First, multiply 2x by x squared minus 2 x minus 3. Then, multiply -1 by x squared minus 2 x minus 3:

2 x open parentheses x squared minus 2 x minus 3 close parentheses equals 2 x cubed minus 4 x squared minus 6 x
short dash 1 open parentheses x squared minus 2 x minus 3 close parentheses equals short dash x squared plus 2 x plus 3

Our final step is to add these two polynomials. Remember to use coefficients of 0 to keep the vertical alignment between like terms.

table attributes columnalign left end attributes row cell table attributes columnalign left end attributes row cell space space space open parentheses 2 x cubed minus 4 x squared minus 6 x plus 0 close parentheses end cell row cell plus stack open parentheses 0 x cubed minus 1 x squared plus 2 x plus 3 close parentheses with bar below end cell end table end cell row cell space space space space space 2 x cubed minus 5 x squared minus 4 x plus 3 end cell end table

summary
When multiplying polynomials, you need to use distribution and multiply each term in the first polynomial by each term in the second polynomial. This is also referred to as FOIL. When multiplying three binomials, you should first multiply or FOIL two factors together and combine any like terms to simplify the polynomial. Then multiply the third factor. Finally, you should write the polynomial in standard form, ordering the terms by the degree from highest to lowest.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

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