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
