1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 1 Chapter 6 Polynomial Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 2 6.5 Factoring Polynomials

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 3 Example: Factor by Grouping Factor 3x 3 – 18x 2 – 2x + 12.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 4 Solution Begin by factoring the first two terms and the last two: We verify the result by finding the product:

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 5 Factoring by Grouping It is a common error to think a polynomial such as 3x 2 (x – 6) – 2(x – 6) is factored. Even though both terms are factored, the entire expression is a difference, not a product. A factored polynomial is a product. Warning

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 6 Factoring by Grouping For a polynomial with four terms, we factor by grouping (if it can be done) by 1. Factoring the first two terms and the last two terms. 2. Factoring out the binomial GCF.

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 7 Example: Factoring by Grouping Factor 10x 3 – 6x 2 + 5x – 3.

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 8 Solution

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 9 Solution Use a graphing calculator to verify our work.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 10 Method 1: Factoring Trinomials by Trial and Error One way to factor trinomials of the form ax 2 + bx + c is to make educated guesses at the factorization and then find the product of these guesses to see if any of them work. This method is called factoring by trial and error.

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 11 Example: Factoring by Trial and Error Factor 2x 2 – 5x – 25.

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 12 Solution If we factor 2x 2 – 5x – 25,the result will be of the form (2x + ?)(x + ?). The product of the last terms must be –25, so the last terms must be 1 and –25, 5 and –5, or –1 and 25, where we an write each pair in either order.

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 13 Solution Therefore, 2x 2 – 5x – 25 = (2x + 5)(x – 5).

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 14 Factoring ax 2 + bx + c by Trial and Error To factor a trinomial of the form ax 2 + bx + c by trial and error, identify possible products using the fact that if the trinomial can be factored as a product of two binomials, then the product of the coefficients of the first terms of the binomials is equal to a and the product of the last terms of the binomials is equal to c.

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 15 Factoring ax 2 + bx + c by Trial and Error For example,

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 16 Factoring ax 2 + bx + c by Trial and Error To find the correct factored expression, multiply the possible products and identify those for which the coefficient of x is b.

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 17 Example: Completely Factoring a Polynomial Factor 6x 3 y 2 + 26x 2 y 3 + 24xy 4.

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 18 Solution First factor out the GCF, 2xy 2 : If we can factor further, the result will be in the form 2xy 2 (3x + ?)(x + ?). The product of the last terms must be 12y 2. Possibilities are shown on the next slide.

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 19 Solution So,

20. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 20 Method 2: Factoring Trinomials by Grouping Instead of using trial and error to factor a trinomial, we can factor by grouping. To factor a trinomial of the form ax 2 + bx + c, look for two integers whose product is ac and whose sum is b.

21. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 21 Factoring ax 2 + bx + c by Grouping To factor a trinomial of the form ax 2 + bx + c by grouping (if it can be done), 1. Find pairs of numbers whose product is ac. 2. Determine which of the pairs of numbers from step 1 has the sum b. Call this pair of numbers m and n. 3. Write the bx term as mx + nx: ax 2 + bx + c = ax 2 + mx + nx + c 4. Factor ax 2 + mx + nx + c by grouping. Another name for this technique is the ac method.

22. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 22 Example: Factoring a Trinomial by Grouping Factor 6x 2 – 7x + 2 by grouping.

23. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 23 Solution Here, a = 6, b = –7, and c = 2. Step 1: Find the product ac: ac = 6(2) = 12. Step 2: Find numbers m and n that have the product ac = 12 and the sum b = –7. So, m and n are –3 and –4. Factor 6x 2 – 7x + 2 by grouping.

24. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.5, Slide 24 Solution Step 3: Write Step 4: Factor by grouping: