Special Pattern Binomials


Math A	More Binomials

The following are special multiplications involving binomials that you will want to try to remember. Be sure to notice the patterns in each situation. You will be seeing these patterns in numerous problems.

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

(a+b)(a-b) = a² - b²

Don't panic! If you cannot remember these patterns, you can arrive at your answer by simply multiplying with FOIL or the vertical method. These patterns are, however, very popular. If you can remember the patterns, you can save yourself some work.

Let's examine these patterns:

Squaring a Binomial - multiplying times itself

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

Notice the middle terms in both of these problems. In each problem the middle term is twice the multiplication of the values used to create the binomial expression.

Example: (x + 3)²	= (x + 3)(x + 3)
	= x² + 3x + 3x + 9 FOIL
	= x² + 6x + 9

* Notice the middle term.

Example: (x - 4)²	= (x-4)(x-4)
	= x² - 4x - 4x + 16 FOIL
	= x² - 8x + 16

* Again, notice the middle term.

Product of Sum and Difference

(notice that the binomials differ only by the sign between the terms)

(a + b)(a - b) = a² - b²

Notice that the middle term disappears. When
multiplication occurs, the values that would form the
middle term actually cancel each other out.

Example: (x + 3)(x - 3)	= x² - 3x + 3x - 9 FOIL
	= x² - 9

*Notice how the middle term disappears.

Example:

(2x + 3y)(2x - 3y)	= 4x² - 6xy + 6xy - 9y² FOIL
	= 4x² - 9y²

* Again, notice how the middle term disappears.

	Roberts