1. Chapter 7 Section 3 and 4 Factoring

2. Objectives  Factor quadratic trinomials of the form x 2 + bx + c. Factor quadratic trinomials of the form ax 2 + bx + c.

3. Factoring  In Chapter 7, you learned how to multiply two binomials using the Distributive Property or the FOIL method. In this lesson, you will learn how to factor a trinomial into two binominals.

4. Factoring  Notice that when you multiply (x + 2)(x + 5), the constant term in the trinomial is the product of the constants in the binomials. (x + 2)(x + 5) = x 2 + 7x + 10 You can use this fact to factor a trinomial into its binomial factors. Look for two numbers that are factors of the constant term in the trinomial. Write two binomials with those numbers, and then multiply to see if you are correct.

5. Example #1  Factor x 2 + 15x + 36 by guess and check.

6. Factoring  The guess and check method is usually not the most efficient method of factoring a trinomial. Look at the product of (x + 3) and (x + 4).  The coefficient of the middle term is the sum of 3 and 4. The third term is the product of 3 and 4. (x + 3)(x +4) = x 2 + 7x + 12

8. Factoring  When c is positive, its factors have the same sign. The sign of b tells you whether the factors are positive or negative. When b is positive, the factors are positive and when b is negative, the factors are negative.

9. Example #2  Factor each trinomial. Check your answer. x 2 + 6x + 5 (x + )(x + ) b = 6 and c = 5; look for factors of 5 whose sum is 6. Factors of 5 Sum 1 and 5 6 (x + 1)(x + 5)

10. Solution  Check (x + 1)(x + 5) = x 2 + x + 5x + 5 = x 2 + 6x + 5

11. Example#3  Factor each trinomial. Check your answer. x 2 + 6x + 9

12. Check it out!!!  Factor each trinomial. Check your answer.  A)  B) x 2 – 8x + 15 x 2 + 8x + 12

13. Factoring  When c is negative, its factors have opposite signs. The sign of b tells you which factor is positive and which is negative. The factor with the greater absolute value has the same sign as b.

14. Example#4  Factor each trinomial. x 2 + x – 20 Factors of –20 Sum –1 and 20 19 –2 and 10 8  –4 and 5 1  (x + )(x + ) b = 1 and c = –20; look for factors of –20 whose sum is 1. The factor with the greater absolute value is positive. (x – 4)(x + 5)

15. Example#5  Factor each trinomial. x 2 – 3x – 18

16. Check it out!!!  Factor each trinomial. Check your answer. x 2 + 2x – 15 x 2 – 6x + 8

17. Factoring  In the previous lesson you factored trinomials of the form x 2 + bx + c. Now you will factor trinomials of the form ax 2 + bx + c, where a ≠ 0.

18. Factoring  When you multiply (3x + 2)(2x + 5), the coefficient of the x 2 -term is the product of the coefficients of the x-terms. Also, the constant term in the trinomial is the product of the constants in the binomials. (3x + 2)(2x + 5) = 6x 2 + 19x + 10

19. Factoring  To factor a trinomial like ax 2 + bx + c into its binomial factors, write two sets of parentheses  ( x + )( x + ). Write two numbers that are factors of a next to the x ’ s and two numbers that are factors of c in the other blanks. Multiply the binomials to see if you are correct. (3x + 2)(2x + 5) = 6x 2 + 19x + 10

20. Example#6  Factor each trinomial by guess and check. 6x 2 + 11x + 3

21. Factoring  So, to factor a 2 + bx + c, check the factors of a and the factors of c in the binomials. The sum of the products of the outer and inner terms should be b. ( X + )( x + ) = ax 2 + bx + c Product = a Product = c Sum of outer and inner products = b

22. Factoring  Factor each trinomial. Check your answer. 2x 2 + 17x + 21 ( x + )( x + ) Factors of 2 Factors of 21 Outer + Inner 1 and 2 1 and 211(21) + 2(1) = 23 1 and 2 21 and 11(1) + 2(21) = 43 1 and 2 3 and 71(7) + 2(3) = 13 1 and 2 7 and 31(3) + 2(7) = 17    (x + 7)(2x + 3)

23. Factoring  Factor each trinomial. Check your answer. 3x 2 – 16x + 16

24. Check it out!!!  Factor each trinomial. Check your answer  A)  B) 9x 2 – 15x + 4 3x 2 + 13x + 12

25. Factioring  When c is negative, one factor of c will be positive and the other factor will be negative. Only some of the factors are shown in the examples, but you may need to check all of the possibilities.

26. Example  Factor each trinomial. Check your answer. 3n 2 + 11n – 4

27. Example  Factor each trinomial. Check your answer. 2x 2 + 9x – 18

28. Check It out !!  Factor each trinomial. Check your answer. 6x 2 + 7x – 3 4n 2 – n – 3

29. Example 4A: Factoring ax 2 + bx + c When a is Negative  Factor –2x 2 – 5x – 3.

30. Student guided practice  Do problems 4,5,6,10 from page 476  Do problems 7,8,9,13 and 19 from page 484

31. Homework  Do problems 20, 21,26,27 from page 476  Do problems 34,35,36,48 and 49 in your book page 484

32. Closure  Today we learned about factorization  Next class we are going to learn about quadratic functions