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Special Products of Binomials

Author: Sophia
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what's covered
In this lesson, you will learn how to expand a special product binomial. Specifically, this lesson will cover:

Table of Contents

1. FOIL Review

Our special products of binomials come through by noticing how expressions are simplified in binomial multiplication. Binomial multiplication is often modeled through a process known as FOIL, which is an acronym that stands for First, Outside, Inside, Last. It describes an order to multiply terms that make up the two factors in binomial multiplication.

EXAMPLE

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

open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses Multiply first terms: x times x equals x squared
x squared Multiply outside terms: x times short dash 3 equals short dash 3 x
x squared minus 3 x Multiply inside terms: 2 times x equals 2 x
x squared minus 3 x plus 2 x Multiply last terms: 2 times short dash 3 equals short dash 6
x squared minus 3 x plus 2 x minus 6 Combine like terms
x squared minus x minus 6 Our solution

2. Square of a Binomial Sum

Take a look at what happens when we expand a binomial sum that is being squared:

EXAMPLE

Multiply open parentheses x plus 3 close parentheses squared.

open parentheses x plus 3 close parentheses squared Rewrite as multiplication of two factors, open parentheses x plus 3 close parentheses
open parentheses x plus 3 close parentheses open parentheses x plus 3 close parentheses FOIL
x squared plus 3 x plus 3 x plus 9 Combine like terms
x squared plus 6 x plus 9 Our solution

Note the relationship between the constant in our binomial sum (3), the coefficient of the x-term (6), and the constant term in the expanded trinomial (9). See if the relationship you spotted holds true in another example:

EXAMPLE

Multiply open parentheses x plus 5 close parentheses squared.

open parentheses x plus 5 close parentheses squared Rewrite as multiplication of two factors, open parentheses x plus 5 close parentheses
open parentheses x plus 5 close parentheses open parentheses x plus 5 close parentheses FOIL
x squared plus 5 x plus 5 x plus 25 Combine like terms
x squared plus 10 x plus 25 Our solution

What is the relationship between 5, 10, and 25? If we take the constant from the binomial sum, 5, and double this number, we get the coefficient of x-term, 10. And if we square 5, we get 25, which is the constant term in the expanded trinomial. This holds true for all squares of binomial sums.

formula to know
Square of a Binomial Sum
left parenthesis x plus a right parenthesis squared space equals space x squared plus 2 a x plus a squared

Test it out with a few more examples:

Original Expression Coefficient of x-term Constant Expanded Expression
open parentheses x plus 2 close parentheses squared 2 times 2 equals 4 2 squared equals 4 x squared plus 4 x plus 4
open parentheses x plus 7 close parentheses squared 2 times 7 equals 14 7 squared equals 49 x squared plus 14 x plus 49
open parentheses x plus 12 close parentheses squared 2 times 12 equals 24 12 squared equals 144 x squared plus 24 x plus 144

3. Square of a Binomial Difference

Next, let's take a look at similar expressions, except these examples involve subtraction instead of addition:

EXAMPLE

Multiply open parentheses x minus 3 close parentheses squared.

open parentheses x minus 3 close parentheses squared Rewrite as multiplication of two factors, open parentheses x minus 3 close parentheses
open parentheses x minus 3 close parentheses open parentheses x minus 3 close parentheses FOIL
x squared minus 3 x minus 3 x plus 9 Combine like terms
x squared minus 6 x plus 9 Our solution

Note that the only distinction here is that the x-term is a negative 6x, because we subtracted 3 from x in the binomial. We still have +9 because -3 times -3 is a positive number. Let's take a look at another example:

EXAMPLE

Multiply open parentheses x minus 5 close parentheses squared.

open parentheses x minus 5 close parentheses squared Rewrite as multiplication of two factors, open parentheses x minus 5 close parentheses
open parentheses x minus 5 close parentheses open parentheses x minus 5 close parentheses FOIL
x squared minus 5 x minus 5 x plus 25 Combine like terms
x squared minus 10 x plus 25 Our solution

We can generalize this as the square of a binomial difference:

formula to know
Square of a Binomial Difference
left parenthesis x minus a right parenthesis squared equals x squared minus 2 a x plus a squared

Test it out with a few more examples:

Original Expression Coefficient of x-term Constant Expanded Expression
open parentheses x minus 4 close parentheses squared 2 times short dash 4 equals short dash 8 open parentheses short dash 4 close parentheses squared equals 16 x squared minus 8 x plus 16
open parentheses x minus 8 close parentheses squared 2 times short dash 8 equals short dash 16 open parentheses short dash 8 close parentheses squared equals 64 x squared minus 16 x plus 64
open parentheses x minus 10 close parentheses squared 2 times short dash 10 equals short dash 20 open parentheses short dash 10 close parentheses squared equals 100 x squared minus 20 x plus 100

big idea
The expanded form for the square of a binomial sum and the square of a binomial difference are also referred to as perfect square trinomials because they consist of three terms, which can be simplified into an expression squared.

term to know
Perfect Square Trinomial
A polynomial with three terms, we can be simplified as a binomial squared, left parenthesis x plus a right parenthesis squared or left parenthesis x minus a right parenthesis squared.

4. Difference of Squares

So far we have discussed special products involving a binomial squared, where a is either positive or negative. We can have a binomial sum open parentheses x plus a close parentheses squared or a binomial difference open parentheses x minus a close parentheses squared. Now, we are going to consider the case where we have a binomial sum multiplied by a binomial difference.

EXAMPLE

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

open parentheses x minus 3 close parentheses open parentheses x plus 3 close parentheses FOIL
x squared plus 3 x minus 3 x minus 9 Combine like terms
x squared minus 9 Our solution

There are a couple of things to note in this example:

  • The x-terms canceled, because the coefficients were opposites of each other.
  • The constant term here is negative, because the product of a positive and negative number is always negative.
  • The constant term is also a perfect square.
We can generalize this as a difference of squares.

formula to know
Difference of Squares
left parenthesis x plus a right parenthesis left parenthesis x minus a right parenthesis equals space x squared minus a squared

Test it out with a few more examples:

Original Expression Coefficient of x-term Constant Expanded Expression
open parentheses x minus 2 close parentheses open parentheses x plus 2 close parentheses short dash 2 plus 2 equals 0 short dash 2 times 2 equals short dash 4 x squared minus 4
open parentheses x minus 7 close parentheses open parentheses x plus 7 close parentheses short dash 7 plus 7 equals 0 short dash 7 times 7 equals short dash 49 x squared minus 49
open parentheses x minus 13 close parentheses open parentheses x plus 13 close parentheses short dash 13 plus 13 equals 0 short dash 13 times 13 equals short dash 169 x squared minus 169

term to know
Difference of Squares
Two squared terms separated by subtraction, x squared minus a squared comma which can be expressed as left parenthesis x plus a right parenthesis left parenthesis x minus a right parenthesis.

summary
A foil review reminds us that FOIL is the acronym used to remember the way to multiply two binomials. It stands for First, Outside, Inside, Last. Recognizing special products of binomials can help make factoring and FOILing easier. The three special products are square of a binomial sum: open parentheses x plus a close parentheses squared comma square of a binomial difference: open parentheses x minus a close parentheses squared comma and the difference of squares: open parentheses x plus a close parentheses open parentheses x minus a close parentheses. A perfect square trinomial is a polynomial with three terms that can be simplified as a binomial squared.

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

Terms to Know
Difference of Squares

Two squared terms separated by subtraction, x squared minus a squared comma which can be expressed as open parentheses x plus a close parentheses open parentheses x minus a close parentheses.

Perfect Square Trinomial

A polynomial with three terms, which can be simplified as a binomial squared, open parentheses x plus a close parentheses squared or open parentheses x minus a close parentheses squared.

Formulas to Know
Difference of Squares

open parentheses x plus a close parentheses open parentheses x minus a close parentheses equals x squared minus a squared

Square of a Binomial Difference

open parentheses x minus a close parentheses squared equals x squared minus 2 a x plus a squared

Square of a Binomial Sum

open parentheses x plus a close parentheses squared equals x squared plus 2 a x plus a squared

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