1. 7.6 Factoring Differences of Squares CORD Math Mrs. Spitz Fall 2006

2. Objectives Identify and factor polynomials that are the differences of squares.

3. Assignment pp. 278-279 #4-40 all Mid-chapter Review pg. 280 Quiz B

4. Application Every Friday in Ms. Spitz’s CORD math class, each student is given a problem that must be solved without using paper or pencil. This week Justin’s problem was to find the product of 93 · 87. How did he determine the answer of 8,091 without doing the multiplication on paper or using a calculator?

5. Application Justin noticed that this product could be written as the product of a sum and a difference.

6. Application He then did the calculation mentally using the rule for this special product.

7. 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.

8. Product of sum and difference Consider the two squares shown below. The area of the larger square is a 2 and the area of the smaller square is b 2. b b a a The area a 2 – b 2 can be found by subtracting the area of the smaller square from the area of the larger square.

9. Difference of Squares a 2 – b 2 = (a – b)(a + b) = (a + b)(a – b)

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

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

12. Ex. 3: Factor You can use this rule to factor trinomials that can be written in the form a 2 – b 2.

13. Ex. 4: Factor 12x 3 – 27xy 2 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. 12x 3 – 27xy 2 = 3x(4x 2 – 9y 2 ) = 3x(2x – 3y)(2x + 3y)

14. Ex. 5: Factor 162m 4 – 32n 8 162m 4 – 32n 8 = 2(81m 4 – 16n 8 ) = 2(9m 2 – 4n 4 )(9m 2 + 4n 4 ) = 2(3m – 2n 2 )(3m + 2n 2 )(9m 2 + 4n 4 ) 9m 2 + 4n 4 cannot be factored because it is not a difference of squares.

15. The measure of a rectangular solid is 5x 3 – 20x + 2x 2 – 8. Find the measures of the dimensions of a solid if each one can be written as a binomial with integral coefficients. 5x 3 – 20x + 2x 2 – 8 = (5x 3 – 20x) + (2x 2 -8) = 5x(x 2 – 4) + 2(x 2 - 4) = (5x+ 2)(x 2 - 4) = (5x + 2)(x – 2)(x + 2) The measures of the dimensions are (5x + 2), (x – 2), and (x + 2).