Factoring Special Products 7-5 Factoring Special Products Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Warm Up Determine whether the following are perfect squares. If so, find the square root. 64 2. 36 yes; 6 3. 45 4. x2 yes; x 5. y8

ESSENTIAL QUESTION How do you factor perfect-square trinomials and the difference of two squares?

Example 1A: Recognizing and Factoring Perfect-Square Trinomials Determine whether each trinomial is a perfect square. If so, factor. If not explain. 9x2 – 15x + 64 9x2 – 15x + 64 8 8 3x 3x 2(3x 8)  2(3x 8) ≠ –15x.  9x2 – 15x + 64 is not a perfect-square trinomial because –15x ≠ 2(3x  8).

Example Determine whether each trinomial is a perfect square. If so, factor. If not explain. Method 2 Use the rule. 81x2 + 90x + 25 a = 9x, b = 5 (9x)2 + 2(9x)(5) + 52 Write the trinomial as a2 + 2ab + b2. Write the trinomial as (a + b)2. (9x + 5)2

Example: Recognizing as Perfect-Square Trinomials 36x2 – 10x + 14 36x2 – 10x + 14 36x2 – 10x + 14 is not a perfect-square trinomial.

Example Determine whether each trinomial is a perfect square. If so, factor. If not explain. Method 1 Factor. x2 + 4x + 4 Factors of 4 Sum (1 and 4) 5 (2 and 2) 4   (x + 2)(x + 2) = (x + 2)2

Example Determine whether each trinomial is a perfect square. If so, factor. If not explain. Method 2 Use the rule. x2 – 14x + 49 a = 1, b = 7 (x)2 – 2(x)(7) + 72 (x – 7)2 Write the trinomial as (a – b)2.

Example: Recognizing and Factoring the Difference of Two Squares 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.

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

Example Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. x4 – 25y6 x4 – 25y6 5y3 5y3  x2 x2 (x2 + 5y3)(x2 – 5y3)

Example 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)

Example 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)

Example 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.

Lesson Quiz: Part I Determine whether each trinomial is a perfect square. If so factor. If not, explain. 64x2 – 40x + 25 2. 121x2 – 44x + 4 3. 49x2 + 140x + 100

Lesson Quiz: Part II Determine whether the binomial is a difference of two squares. If so, factor. If not, explain. 5. 9x2 – y4 6. 30x2 – y2 7. x2 – y8