Lesson Objectives: I will be able to … Choose an appropriate method for factoring a polynomial Combine methods for factoring a polynomial Language Objective: I will be able to … Read, write, and listen about vocabulary, key concepts, and examples

Not Fully Factored Example: 3x(2x + 4) Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further. Fully Factored Example: 4x2(3x – 2) Not Fully Factored Example: 3x(2x + 4) (2 can also be factored from 2x and 4.)

Example 1: Determining Whether a Polynomial is Completely Factored Page 23 Tell whether each polynomial is completely factored. If not factor it. A. 3x2(6x – 4) 3x2(6x – 4) 6x – 4 can be further factored. 6x2(3x – 2) Factor out 2, the GCF of 6x and – 4. 6x2(3x – 2) is completely factored. B. (x2 + 1)(x – 5) (x2 + 1)(x – 5) Neither x2 +1 nor x – 5 can be factored further. (x2 + 1)(x – 5) is completely factored.

x2 + 4 is a sum of squares, and cannot be factored. Caution

Tell whether the polynomial is completely factored. If not, factor it. Your Turn 1 Page 24 Tell whether the polynomial is completely factored. If not, factor it. A. 5x2(x – 1) Neither 5x2 nor x – 1 can be factored further. 5x2(x – 1) is completely factored. B. (4x + 4)(x + 1) (4x + 4)(x + 1) 4x + 4 can be further factored. 4(x + 1)(x + 1) Factor out 4, the GCF of 4x and 4. 4(x + 1)2 is completely factored.

To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below to factor a polynomial completely. Page 22

Example 2: Factoring by GCF and Recognizing Patterns Page 24 Factor the polynomial completely. Check your answer. 10x2 + 48x + 32 2(5x2 + 24x + 16) Factor out the GCF. 2(5x + 4)(x + 4) Factor remaining trinomial. Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16) = 10x2 + 40x + 8x + 32 = 10x2 + 48x + 32 

Example 3: Factoring by GCF and Recognizing Patterns Page 25 Factor the polynomial completely. Check your answer. 8x6y2 – 18x2y2 Factor out the GCF. 4x4 – 9 is a perfect-square trinomial of the form a2 – b2. 2x2y2(4x4 – 9) 2x2y2(2x2 – 3)(2x2 + 3) a = 2x, b = 3 Check 2x2y2(2x2 – 3)(2x2 + 3) = 2x2y2(4x4 – 9) = 8x6y2 – 18x2y2 

Factor the polynomial completely. Check your answer. Your Turn 3 Page 25 Factor the polynomial completely. Check your answer. 4x3 + 16x2 + 16x 4x(x2 + 4x + 4) Factor out the GCF. Factor out the GCF. x2 + 4x + 4 is a perfect-square trinomial of the form a2 + 2ab + b2. a = x, b = 2 4x(x + 2)2 Check 4x(x + 2)2 = 4x(x2 + 2x + 2x + 4) = 4x(x2 + 4x + 4) = 4x3 + 16x2 + 16x 

If none of the factoring methods work, the polynomial is said to be unfactorable. For a polynomial of the form ax2 + bx + c, if there are no numbers whose sum is b and whose product is ac, then the polynomial is unfactorable. In other words, if the diamond problem is unsolvable, then the polynomial is unfactorable. Helpful Hint

Example 4: Factoring by Multiple Methods Page 26 Factor each polynomial completely. 9x2 + 3x – 2 The GCF is 1 and there is no pattern. ( x + )( x + ) a = 9 and c = –2; Outer + Inner = 3 Factors of 9 Factors of 2 Outer + Inner 1 and 9 1 and –2 1(–2) + 1(9) = 7  3 and 3 1 and –2 3(–2) + 1(3) = –3   3 and 3 –1 and 2 3(2) + 3(–1) = 3 (3x – 1)(3x + 2)

Example 5: Factoring by Multiple Methods Page 26 Factor the polynomial completely. 12b3 + 48b2 + 48b The GCF is 12b; (b2 + 4b + 4) is a perfect-square trinomial in the form of a2 + 2ab + b2. 12b(b2 + 4b + 4) (b + )(b + ) Factors of 4 Sum 1 and 4 5  2 and 2 4  a = 2 and c = 2 12b(b + 2)(b + 2) 12b(b + 2)2

Example 6: Factoring by Multiple Methods Page 27 Factor the polynomial completely. 4y2 + 12y – 72 Factor out the GCF. There is no pattern. b = 3 and c = –18; look for factors of –18 whose sum is 3. 4(y2 + 3y – 18) (y + )(y + ) Factors of –18 Sum –1 and 18 17  –2 and 9 7   –3 and 6 3 The factors needed are –3 and 6 4(y – 3)(y + 6)

Example 7: Factoring by Multiple Methods. Page 27 Factor the polynomial completely. (x4 – x2) x2(x2 – 1) Factor out the GCF. x2(x + 1)(x – 1) x2 – 1 is a difference of two squares.

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 Your Turn 7 Factor the polynomial completely. 2p5 + 10p4 – 12p3 Page 28 Factor the polynomial completely. 2p5 + 10p4 – 12p3 Factor out the GCF. There is no pattern. b = 5 and c = –6; look for factors of –6 whose sum is 5. 2p3(p2 + 5p – 6) (p + )(p + ) Factors of – 6 Sum  The factors needed are –1 and 6 – 1 and 6 5 2p3(p + 6)(p – 1)

Chapter 8 Quick Review Assignment #16 1. Write the prime factorization of 128. 2. Factor out the GCF: 6x5y2 – 15xy3 3. Factor by grouping: 6a3 – 4a2 – 15a + 10 4. Factor each trinomial. a) x2 – 11x + 28 b) 10x2 – 31x + 15 5. Determine whether each polynomial is a perfect square trinomial, difference of two squares, or neither. If it is a special product, factor it. If not, explain why. a) 4 – 16x4 b) 9x2 + 6x + 1 6. Factor completely: -18d2 – 3d + 6

Chapter 8 Group Quiz next class! Homework Assignment #16 Holt 8-6 #19 – 30, 40 – 45 8-6 Practice C Worksheet Chapter 8 Group Quiz next class!