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Multiplying Binomials

Multiplying Binomials

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In this lesson, students will learn how to multiply binomials by using the FOIL method.

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Tutorial
This tutorial covers multiplying binomials, through the definition and discussion of:
  1. The Distributive Property: A Review
  2. Multiplying Binomials Using FOIL
  3. Multiplying Binomials Squared Using FOIL


1. The Distributive Property: A Review

In review, the distributive property is used when simplifying an expression such as:

8 x squared left parenthesis 2 x cubed minus 1 right parenthesis

Using the distributive property, you would distribute the outside expression, 8x^2, to both terms inside the parentheses. Therefore, you would first multiply 8x^2 by the first term, noting that because both of the bases are x, you can add your exponents. Next, multiply 8x^2 by your second term, -1. Thus, your final answer is:

table attributes columnalign left end attributes row cell 8 x squared left parenthesis 2 x cubed minus 1 right parenthesis equals end cell row cell 16 x to the power of 5 minus 8 x squared end cell end table

Your answer is in standard form, or descending order according to the exponents in each term.


2. Multiplying Binomials Using FOIL

Suppose you want to multiply two binomials:

left parenthesis 3 x plus 3 right parenthesis left parenthesis 2 x minus 4 right parenthesis

To multiply binomials, you need to distribute twice, multiplying the 3x by both terms in the second parentheses, following by multiplying the 3 by both terms in the second parentheses. Note that 3x and 2x both have implied exponents of 1.

table attributes columnalign left end attributes row cell left parenthesis 3 x plus 3 right parenthesis left parenthesis 2 x minus 4 right parenthesis equals end cell row cell 6 x squared minus 12 x plus 6 x minus 12 end cell end table

When distributing, you will write all terms as a single expression—adding terms with positive coefficients and subtracting terms with negative coefficients. Therefore, you can combine the like terms, -12x and 6x, which equals -6x, providing your final answer:

6 x squared minus 6 x minus 12
KEY FORMULA
(a+b)(c+d) = ac+ad+bc+bd
Try using the distributive property to multiply these two binomials:
left parenthesis 5 x minus 3 right parenthesis left parenthesis x plus 2 right parenthesis

5x times x will give you 5x^2, and 5x times 2 equals 10x. Then, -3 times x equals -3x, and -3 times 2 equals -6.

table attributes columnalign left end attributes row cell left parenthesis 5 x minus 3 right parenthesis left parenthesis x plus 2 right parenthesis equals end cell row cell 5 x squared plus 10 x minus 3 x minus 6 end cell end table

Finally, combine the like terms, 10x and -3x, to arrive at your final answer:

5 x squared plus 7 x minus 6

You can remember this method of multiplying binomials by using the acronym “FOIL,” which stands for “First, Outside, Inside, and Last.” In the preceding example, you distributed by multiplying the first two terms in each parentheses and then the outside terms. You then distributed by multiplying the inside terms and then the last two terms in each parentheses. Therefore, you can use the acronym FOIL to remember the steps for distributing factors in binomial multiplication.

File:1183-foil1.PNG

FOIL
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last

IN CONTEXT

Suppose a farmer wants to plant a small area for a new chicken pen. The length and width of the pen are shown below. What is the area of the chicken pen in terms of x?

File:1184-foil2.PNG

To find the area of the pen, you want to multiply the length and the width, which would be expressed as:
left parenthesis x plus 3 right parenthesis left parenthesis x plus 10 right parenthesis
These are binomials multiplied together, so you can multiply using FOIL:

F Your first two terms, x and x, multiply together to give you x^2.
O Your outside terms, x times 10, equal 10x.
I Multiplying your inside terms, 3 and x, equals 3x.
L Lastly, multiplying your last terms, 3 and 10, equals 30.
x squared plus 10 x plus 3 x plus 30
You can combine your like terms, 10x and 3x. Therefore, the area of the chicken pen can be written as:
x squared plus 13 x plus 30
Use FOIL to multiply the following two binomials. Note that one of these binomials includes an exponent.
left parenthesis 2 x squared plus 4 x right parenthesis left parenthesis 3 x plus 2 right parenthesis
F Multiply your first terms together, 2x^2 times 3x.
O Multiply your outside terms, 2x^2 times 2.
I Multiply your inside terms, 4x times 3x.
L Multiply your last terms, 4x times 2.

Combine your like terms to provide your final expression:

table attributes columnalign left end attributes row cell left parenthesis 2 x squared plus 4 x right parenthesis left parenthesis 3 x plus 2 right parenthesis equals end cell row cell 6 x cubed plus 4 x squared plus 12 x squared plus 8 x equals end cell row cell 6 x cubed plus 16 x squared plus 8 x end cell end table


3. Multiplying Binomials Squared Using FOIL

Suppose you want to simplify:

left parenthesis x minus 5 right parenthesis squared

This is an example of a binomial squared, and it means the same as:

left parenthesis x minus 5 right parenthesis left parenthesis x minus 5 right parenthesis

You can multiply binomials squared in the same manner as other binomials, using FOIL: multiply your first two terms, your outside terms, your inside terms and finally, your last terms:

table attributes columnalign left end attributes row cell left parenthesis x minus 5 right parenthesis left parenthesis x minus 5 right parenthesis equals end cell row cell x squared minus 5 x minus 5 x plus 25 end cell end table

Combine your like terms, -5x and -5x, to provide your final expression:

x squared minus 10 x plus 25

Today you reviewed the distributive property and how to use it when multiplying binomials. You learned about the acronym FOIL, which is used to remember the steps for distributing factors in binomial multiplication: First, Outside, Inside, and Last. Lastly, you learned how to use foil when multiplying binomials squared.

Source: This work is adapted from Sophia author Colleen Atakpu.

TERMS TO KNOW
  • KEY FORMULA

    (a+b)(c+d) = ac+ad+bc+bd

  • FOIL

    An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last.