7.6 Factoring Differences of Squares CORD Math Mrs. Spitz Fall 2006

Product of sum and difference The product of the sum and difference of two expressions is called “the difference of squares.” The process for finding this product can be reversed in order to factor the difference of squares. Factoring the difference of squares can be modeled geometrically.

Difference of Squares a2 – b2 = (a – b)(a + b) = (a + b)(a – b)

Ex. 1: Factor a2 - 64 a2 – 64 = (a)2 – (8)2 = (a – 8)(a + 8) You can use this rule to factor trinomials that can be written in the form a2 – b2. a2 – 64 = (a)2 – (8)2 = (a – 8)(a + 8)

Ex. 2: Factor 9x2 – 100y2 9x2 – 100y2 = (3x)2 – (10y)2 You can use this rule to factor trinomials that can be written in the form a2 – b2. 9x2 – 100y2 = (3x)2 – (10y)2 = (3x – 10y)(3x + 10y)

Ex. 3: Factor You can use this rule to factor trinomials that can be written in the form a2 – b2.

Ex. 4: Factor 12x3 – 27xy2 12x3 – 27xy2 = 3x(4x2 – 9y2) Sometimes the terms of a binomial have common factors. If so, the GCF should always be factored out first. Occasionally, the difference of squares needs to be applied more than once or along with grouping in order to completely factor a polynomial. 12x3 – 27xy2 = 3x(4x2 – 9y2) = 3x(2x – 3y)(2x + 3y)

Ex. 5: Factor 162m4 – 32n8 162m4 – 32n8 = 2(81m4 – 16n8) 9m2 + 4n4 cannot be factored because it is not a difference of squares.

The measure of a rectangular solid is 5x3 – 20x + 2x2 – 8 The measure of a rectangular solid is 5x3 – 20x + 2x2 – 8. Find the measures of the dimensions of a solid if each one can be written as a binomial with integral coefficients. 5x3 – 20x + 2x2 – 8 = (5x3 – 20x) + (2x2 -8) = 5x(x2 – 4) + 2(x2 - 4) = (5x+ 2)(x2 - 4) = (5x + 2)(x – 2)(x + 2) The measures of the dimensions are (5x + 2), (x – 2), and (x + 2).