Multiplying Binomials
Using Patterns to Multiply Two Binomials
In some cases, we can multiply two binomials more quickly by
recognizing a pattern.
Formula — To Square a Binomial
(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2
(a - b)2 = (a - b)(a - b) = a2 - 2ab + b2
Note that (a + b) 2 is NOT equal to a 2 + b 2 .
Don’t forget the middle term, 2ab.
| (a + b)2 |
= (a + b)(a + b) = a2 + ab + ab + b2
= a2 + 2ab + b2 |
The square of a binomial is called a perfect square trinomial.
Both a2 + 2ab + b2 and a2 - 2ab + b2 are perfect square trinomials.
Example 1
Find: (6w - 8)2
Solution
| The expression (6w - 8)2 is
the square of a binomial. Substitute 6w for a and 8 for b.
Simplify. |
(a - b)2 (6w - 8)2 |
= a2 - 2ab + b2 =
(6w)2 - 2(6w)(8) + (8)2
= 36w2 - 96w + 64 |
So, squaring the binomial results in 36w 2 - 96w + 64.
Formula — The Product of the Sum and Difference of the Same Two Terms
(a + b)(a - b) = a2 - b2
The expression a2 - b2 is called a difference of two squares.
Example 2
Find: (5x + y)(5x - y)
Solution
| The expression (5x + y)(5x - y) is the
product of the sum and difference of
the same two terms.
Substitute 5x for a and y for b.
Simplify. |
(a + b)(a - b) (5x + y)(5x
- y)
|
= a2 - b2 = (5x)2
- (y)2
= 25x2 - y2 |
Thus, the product is 25x2 - y2.
|