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Special Products

After studying this lesson, you will be able to:

  • Use Special Products Rules to multiply certain polynomials.

We will consider three special products in this section.

 

Square of a Sum

(a + b) 2 = a 2 + 2ab + b 2

 

Example 1

(x + 3) 2

We are squaring a sum. We can just write the binomial down twice and multiply using the FOIL Method or we can use the Square of a Sum Rule.

Using the Square of a Sum Rule, we:

square the first term which is x...this will give us x 2

multiply the 2 terms together and double them x times 3 is 3x... double it to get 6x

square the last term which is 3...this will give us 9

The answer is x 2 + 6x + 9

 

Example 2

(x + 2) 2

We are squaring a sum. We can just write the binomial down twice and multiply using the FOIL Method or we can use the Square of a Sum Rule.

Using the Square of a Sum Rule, we:

square the first term which is x...this will give us x 2

multiply the 2 terms together and double them x times 2 is 2x...

double it to get 4x square the last term which is 2...this will give us 4

The answer is x 2 + 4x + 4

 

Square of a Difference

(a - b) 2 = a 2 - 2ab + b 2

 

Example 3

(x - 2) 2

We are squaring a difference. We can just write the binomial down twice and multiply using the FOIL Method or we can use the Square of a Sum Rule.

Using the Square of a Difference Rule, we:

square the first term which is x...this will give us x 2

multiply the 2 terms together and double them x times -2 is 2x...double it to get -4x

square the last term which is -2...this will give us 4

The answer is x 2 - 4x + 4

 

Product of a Sum and a Difference

(a + b)(a - b) = a 2 - b 2

 

Example 4

( x + 5 ) ( x - 5 )

We have the product of a sum and a difference. Here's what we do:

multiply the first terms x times x will be x 2

multiply the last terms 5 times 5 will be -25

The answer is x 2 -25

 

Example 5

( x + 7 ) ( x - 7 )

We have the product of a sum and a difference. Here's what we do:

multiply the first terms x times x will be x 2

multiply the last terms 7 times 7 will be - 49

The answer is x 2 - 49

 
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