Polynomials
Multiplication of Binomials -- Special Cases
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) | |
| = | a 2 + ab + ba + b 2 | ||
| = | a 2 + ab + ab + b 2 | ||
| = | a 2 +2ab + b 2 |
The square of a binomial is always the sum of:
- The first term squared,
- 2 times the product of the first and second terms, and
- the second term squared.
When a binomial is squared, the resulting trinomial is called a perfect square trinomial.
Examples:
(x + 5)2 = x
2 +2(x)(5) + 52 = x
2 + 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 = 4x
2 -12xy + 9y
2
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) | |
| = | a 2 - ab + ab - b 2 | ||
| = | a 2 - b 2 |
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) = x
2 -92 = x
2 - 81
(2x - y)(2x + y) = (2x)2 - y
2 = 4x
2 - y
2
(3x
2 -2)(3x
2 +2) = (3x
2)2 -22 = 9x
4 - 4
(- y + 5x)(- y - 5x) = (- y)2 - (5x)2 = y
2 -15x
2
