Factoring the Difference of Algebra 1 ~ Chapter 9.5 Factoring the Difference of Two Perfect Squares

A polynomial is a difference of two squares if: There are two terms, one subtracted from the other. Both terms are perfect squares. 4x2 – 9 2x 2x 3 3 

To Factor the difference of perfect squares, take the square root of each term. In one set of parenthesis the sign is positive, and in the other it is negative.

Recognize a difference of two squares: -the coefficients of variable terms are perfect squares -powers on variable terms are even -constants are perfect squares. Reading Math

Example 1: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 3p2 – 9q4 3p2 – 9q4 3q2  3q2 3p2 is not a perfect square. 3p2 – 9q4 is not the difference of two squares because 3p2 is not a perfect square.

Example 2: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 100x2 – 4y2 100x2 – 4y2 2y 2y  10x 10x The polynomial is a difference of two squares. (10x + 2y)(10x – 2y) Write the polynomial as (a + b)(a – b). 100x2 – 4y2 = (10x + 2y)(10x – 2y)

Example 3: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. x4 – 25y6 x4 – 25y6 5y3 5y3  x2 x2 The polynomial is a difference of two squares. Write the polynomial as (a + b)(a – b). (x2 + 5y3)(x2 – 5y3) x4 – 25y6 = (x2 + 5y3)(x2 – 5y3)

Check It Out! You try it… Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 1 – 4x2 1 – 4x2 2x 2x  1 1 (1 + 2x)(1 – 2x) 1 – 4x2 = (1 + 2x)(1 – 2x)

Check It Out! You try another one… Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. p8 – 49q6 p8 – 49q6 7q3 7q3 ● p4 p4 (p4 + 7q3)(p4 – 7q3) p8 – 49q6 = (p4 + 7q3)(p4 – 7q3)

Check It Out! Try one more… Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 16x2 – 4y5 16x2 – 4y5 4x  4x 16x2 – 4y5 is not the difference of two squares because 4y5 is not a perfect square.

Example 4: Combining with GCF Factor the binomial completely. 48a3 – 12a 12a(4a2 – 1) 12a(2a -1)(2a +1)

Example 4b: You try one… Factor the binomial completely. 3b3 – 27b

Example 5: Doing more than one technique Factor the binomial completely. 2x4 – 162 2(x4 – 81) 2(x2 -9)(x2 +9) 2(x - 3)(x + 3)(x2 +9)

Example 5b: You try one… Factor the binomial completely. 4y4 – 2500

Example 6: Combining Factoring by grouping Factor the binomial completely. 5x3 +15x2 – 5x - 15

Example 6: Your turn… Factor the binomial completely. 6x3 +30x2 – 24x - 120

Example 7: Solving Equations Solve the following: p2 – 25 = 0

Example 7b: Solving Equations Solve the following: 18x3 = 50x

Example 7c: You try one… Solve the following: 48y3 = 3y