8.6 Choosing a Factoring Method

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8.6 Choosing a Factoring Method

California Standards 11.0 Students apply basic factoring techniques to second- and simple third- degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

Check 1. GCF 2. Difference of two squares (2 terms) 3. Factor by grouping (4 terms) 4. X method (x2 + 8x + 12) 5. Perfect squares 6. x and box method (4x2 + 16x + 15)

Tell whether each expression is completely factored. If not, factor it. (x2 + 1)(x – 5) 1. GCF 2. Difference of two squares (2 terms) 3. Factor by grouping (4 terms) 4. X method (x2 + 8x + 12) 5. Perfect squares 6. x and box method (4x2 + 16x + 15) Yes

B. 5x(x2 – 2x – 3) 5x(x2 – 2x – 3) 5x(x + 1)(x – 3) 5x(x + 1)(x – 3) Tell whether each expression is completely factored. If not, factor it. B. 5x(x2 – 2x – 3) 1. GCF 2. Difference of two squares (2 terms) 3. Factor by grouping (4 terms) 4. X method (x2 + 8x + 12) 5. Perfect squares 6. x and box method (4x2 + 16x + 15) 5x(x2 – 2x – 3) 5x(x + 1)(x – 3) 5x(x + 1)(x – 3) Product -3 -3 1 -2 sum

C. 8x6y2 – 18x2y2 Not done 2x2y2(2x2 – 3)(2x2 + 3) 8x6y2 – 18x2y2 2 Tell whether each expression is completely factored. If not, factor it. C. 8x6y2 – 18x2y2 1. GCF 2. Difference of two squares (2 terms) 3. Factor by grouping (4 terms) 4. X method (x2 + 8x + 12) 5. Perfect squares 6. x and box method (4x2 + 16x + 15) 8x6y2 – 18x2y2 2 4x6y2 – 9x2y2 x2 4x4y2 – 9y2 y2 4x4 – 9 Not done 2x2y2(4x4 – 9) 2x2y2(2x2 – 3)(2x2 + 3)

D. 2x2y – 2y3 Not done 2x2y – 2y3 2 x2y – y3 y x2 – y2 2y(x2 – y2) Tell whether each expression is completely factored. If not, factor it. D. 2x2y – 2y3 1. GCF 2. Difference of two squares (2 terms) 3. Factor by grouping (4 terms) 4. X method (x2 + 8x + 12) 5. Perfect squares 6. x and box method (4x2 + 16x + 15) 2x2y – 2y3 2 x2y – y3 y x2 – y2 Not done 2y(x2 – y2) 2y(x – y)(x + y)

E. 12b3 + 48b2 + 48b Product sum 12b3 + 48b2 + 48b 12 b3 + 4b2 + 4b b Tell whether each expression is completely factored. If not, factor it. E. 12b3 + 48b2 + 48b 12b3 + 48b2 + 48b 1. GCF 2. Difference of two squares (2 terms) 3. Factor by grouping (4 terms) 4. X method (x2 + 8x + 12) 5. Perfect squares 6. x and box method (4x2 + 16x + 15) 12 b3 + 4b2 + 4b b b2 + 4b + 4 12b(b2 + 4b + 4) Product 12b(b + 2)(b+ 2) 4 12b(b + 2)2 2 2 4 sum

Factor completely 2p3(p + 6)(p – 1) 2p5 + 10p4 – 12p3 4x3 + 16x2 + 16x 4y2 + 12y – 72 2x2 + 20x + 32 9q3 + 30q2 + 24q 4x(x + 2)2 4(y – 3)(y + 6) 2(x + 8)(x + 2) 3q(3q + 4)(q + 2)

Lesson Quiz Tell whether the polynomial is completely factored. If not, factor it. 1. (x + 3)(5x + 10) 2. 3x2(x2 + 9) no; 5(x+ 3)(x + 2) completely factored Factor each polynomial completely. Check your answer. 3. x3 + 4x2 + 3x + 12 4. 4x2 + 16x – 48 4(x + 6)(x – 2) (x + 4)(x2 + 3) 5. 18x2 – 3x – 3 6. 18x2 – 50y2 3(3x + 1)(2x – 1) 2(3x + 5y)(3x – 5y) 7. 5x – 20x3 + 7 – 28x2 (1 + 2x)(1 – 2x)(5x + 7)