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Objectives Factor the difference of two squares.

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**A polynomial is a difference of two squares if:**

The difference of two squares has the form a2 – b2. The difference of two squares can be written as the product (a + b)(a – b). You can use this pattern to factor some polynomials. 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

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**Recognize a difference of two squares:**

the coefficients of variable terms are perfect squares powers on variable terms are even, and constants are perfect squares. Reading Math

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**Example 3A: 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.

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**Example 3B: 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)2 – (2y)2 a = 10x, b = 2y (10x + 2y)(10x – 2y) Write the polynomial as (a + b)(a – b). 100x2 – 4y2 = (10x + 2y)(10x – 2y)

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**Example 3C: 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. (x2)2 – (5y3)2 a = x2, b = 5y3 Write the polynomial as (a + b)(a – b). (x2 + 5y3)(x2 – 5y3) x4 – 25y6 = (x2 + 5y3)(x2 – 5y3)

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**The polynomial is a difference of two squares.**

Check It Out! Example 3a Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 1 – 4x2 1 – 4x2 2x 2x The polynomial is a difference of two squares. (1) – (2x)2 a = 1, b = 2x (1 + 2x)(1 – 2x) Write the polynomial as (a + b)(a – b). 1 – 4x2 = (1 + 2x)(1 – 2x)

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**The polynomial is a difference of two squares.**

Check It Out! Example 3b Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. p8 – 49q6 p8 – 49q6 7q3 7q3 ● p4 p4 The polynomial is a difference of two squares. (p4)2 – (7q3)2 a = p4, b = 7q3 (p4 + 7q3)(p4 – 7q3) Write the polynomial as (a + b)(a – b). p8 – 49q6 = (p4 + 7q3)(p4 – 7q3)

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Check It Out! Example 3c Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 16x2 – 4y5 16x2 – 4y5 4x 4x 4y5 is not a perfect square. 16x2 – 4y5 is not the difference of two squares because 4y5 is not a perfect square.

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Lesson Quiz: Part II Determine whether the binomial is a difference of two squares. If so, factor. If not, explain. 5. 9x2 – 144y4 6. 30x2 – 64y2 7. 121x2 – 4y8 (3x + 12y2)(3x – 12y2) Not a difference of two squares; 30x2 is not a perfect square (11x + 2y4)(11x – 2y4)

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