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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