### Square of a Binomial

To square a binomial, multiply the binomial by itself:
(a + b)2 = (a + b)(a + b)

 (a + b)2 = (a + b)(a + b) = a2 + ab + ba + b2 = a2 + ab + ab + b2 = a2 +2ab + b2

The square of a binomial is always the sum of:

1. The first term squared,
2. 2 times the product of the first and second terms, and
3. the second term squared.

When a binomial is squared, the resulting trinomial is called a perfect square trinomial.

Examples:

(x + 5)2 = x2 +2(x)(5) + 52 = x2 + 10x + 25
(100 - 1)2 = 1002 +2(100)(- 1) + (- 1)2 = 10000 - 200 + 1 = 9801
(2x - 3y)2 = (2x)2 +2(2x)(- 3y) + (- 3y)2 = 4x2 -12xy + 9y2

### Product of the Sum and Difference of Two Terms

When we multiply two polynomials that are the sum and difference of the same 2 terms -- (x + 5) and (x - 5) for example -- we get an interesting result:

 (a + b)(a - b) = a(a) + a(- b) + ba + b(- b) = a2 - ab + ab - b2 = a2 - b2

The product of the sum and difference of the same two terms is always the difference of two squares; it is the first term squared minus the second term squared. Thus, this resulting binomial is called a difference of squares.

Examples:

(7 - 2)(7 + 2) = 72 -22 = 49 - 4 = 45
(x + 9)(x - 9) = x2 -92 = x2 - 81
(2x - y)(2x + y) = (2x)2 - y2 = 4x2 - y2
(3x2 -2)(3x2 +2) = (3x2)2 -22 = 9x4 - 4
(- y + 5x)(- y - 5x) = (- y)2 - (5x)2 = y2 -15x2