1. 8-4 Factoring ax 2 + bx + c Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 8-4 Factoring ax 2 + bx + c Warm Up Find each product. 1. (x – 2)(2x + 7) 2. (3y + 4)(2y + 9) 3. (3n – 5)(n – 7) Find each trinomial. 4. x 2 + 4x – 32 5. z 2 + 15z + 36 6. h 2 – 17h + 72 6y 2 + 35y + 36 2x 2 + 3x – 14 3n 2 – 26n + 35 (z + 3)(z + 12) (x – 4)(x + 8) (h – 8)(h – 9)

3. 8-4 Factoring ax 2 + bx + c 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. California Standards

4. 8-4 Factoring ax 2 + bx + c 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 or 1.

5. 8-4 Factoring ax 2 + bx + c 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

6. 8-4 Factoring ax 2 + bx + c To factor a trinomial like ax 2 + bx + c into its binomial factors, first write two sets of parentheses: ( x + )( x + ). Write two integers that are factors of a next to the x ’ s and two integers that are factors of c in the other blanks. Then multiply to see if the product is the original trinomial. If there are no two such integers, we say the trinomial is not factorable.

7. 8-4 Factoring ax 2 + bx + c Additional Example 1: Factoring ax 2 + bx + c Factor 6x 2 + 11x + 4. Check your answer. The first term is 6x 2, so at least one variable term has a coefficient other than 1. ( x + )( x + ) The coefficient of the x 2 term is 6. The constant term in the trinomial is 4. Try integer factors of 6 for the coefficients and integer factors of 4 for the constant terms. (1x + 4)(6x + 1) = 6x 2 + 25x + 4  (1x + 2)(6x + 2) = 6x 2 + 14x + 4  (1x + 1)(6x + 4) = 6x 2 + 10x + 4  (2x + 4)(3x + 1) = 6x 2 + 14x + 4  (3x + 4)(2x + 1) = 6x 2 + 11x + 4

8. 8-4 Factoring ax 2 + bx + c Additional Example 1 Continued Factor 6x 2 + 11x + 4. Check your answer. 6x 2 + 11x + 4 = (3x + 4)(2x + 1) The factors of 6x 2 + 11x + 4 are (3x + 4) and (2x + 1). Check (3x + 4)(2x + 1) = Use the FOIL method. 6x 2 + 3x + 8x + 4 = 6x 2 + 11x + 4 The product of the original trinomial.

9. 8-4 Factoring ax 2 + bx + c Check It Out! Example 1a Factor each trinomial. Check your answer. The first term is 6x 2, so at least one variable term has a coefficient other than 1. ( x + )( x + ) The coefficient of the x 2 term is 6. The constant term in the trinomial is 3. 6x 2 + 11x + 3 (1x + 3)(6x + 1) = 6x 2 + 19x + 3  (1x + 1)(6x + 3) = 6x 2 + 9x + 3  Try integer factors of 6 for the coefficients and integer factors of 3 for the constant terms. (2x + 1)(3x + 3) = 6x 2 + 9x + 3  (3x + 1)(2x + 3) = 6x 2 + 11x + 3

10. 8-4 Factoring ax 2 + bx + c Check It Out! Example 1a Continued Factor each trinomial. Check your answer. The factors of 6x 2 + 11x + 3 are (3x + 1)(2x + 3). 6x 2 + 11x + 3 = (3x + 1)(2x +3) Check (3x + 1)(2x + 3) = Use the FOIL method. 6x 2 + 9x + 2x + 3 = 6x 2 + 11x + 3 The product of the original trinomial.

11. 8-4 Factoring ax 2 + bx + c Check It Out! Example 1b Factor each trinomial. Check your answer. The first term is 3x 2, so at least one variable term has a coefficient other than 1. ( x + )( x + ) 3x 2 – 2x – 8 The coefficient of the x 2 term is 3. The constant term in the trinomial is – 8. Try integer factors of 3 for the coefficients and integer factors of 8 for the constant terms. (1x – 1)(3x + 8) = 3x 2 + 5x – 8  (1x – 8)(3x + 1) = 3x 2 – 23x – 8   (1x – 4)(3x + 2) = 3x 2 – 10x – 8 (1x – 2)(3x + 4) = 3x 2 – 2x – 8

12. 8-4 Factoring ax 2 + bx + c Check It Out! Example 1b Factor each trinomial. Check your answer. 3x 2 – 2x – 8 The factors of 3x 2 – 2x – 8 are (x – 2)(3x + 4). 3x 2 – 2x – 8 = (x – 2)(3x + 4) Check (x – 2)(3x + 4) = Use the FOIL method. 3x 2 + 4x – 6x – 8 = 3x 2 – 2x – 8 The product of the original trinomial.

13. 8-4 Factoring ax 2 + bx + c So, to factor ax 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 Sum of outer and inner products = b Product = c Product = a

14. 8-4 Factoring ax 2 + bx + c Since you need to check all the factors of a and all the factors of c, it may be helpful to make a table. Then check the products of the outer and inner terms to see if the sum is b. You can multiply the binomials to check your answer. ( X + )( x + ) = ax 2 + bx + c Sum of outer and inner products = b Product = c Product = a

15. 8-4 Factoring ax 2 + bx + c Additional Example 2A: Factoring ax 2 + bx + c When c is Positive Factor each trinomial. Check your answer. 2x 2 + 17x + 21 ( x + )( x + ) a = 2 and c = 21; Outer + Inner = 17. (x + 7)(2x + 3) 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    Check (x + 7)(2x + 3) = 2x 2 + 3x + 14x + 21 = 2x 2 + 17x + 21 Use the FOILmethod.

16. 8-4 Factoring ax 2 + bx + c When b is negative and c is positive, the factors of c are both negative. Remember!

17. 8-4 Factoring ax 2 + bx + c Factor each trinomial. Check your answer. 3x 2 – 16x + 16 a = 3 and c = 16, Outer + Inner = –16. (x – 4)(3x – 4) Check (x – 4)(3x – 4) = 3x 2 – 4x – 12x + 16 = 3x 2 – 16x + 16 Use the FOIL method. Factors of 3 Factors of 16 Outer + Inner 1 and 3 –1 and –161(–16) + 3(–1) = –19 1 and 3 – 2 and – 8 1( – 8) + 3(–2) = –14 1 and 3 – 4 and – 41( – 4) + 3(– 4)= –16   ( x + )( x + ) Additional Example 2B: Factoring ax 2 + bx + c When c is Positive

18. 8-4 Factoring ax 2 + bx + c Check It Out! Example 2a Factor each trinomial. Check your answer. 6x 2 + 17x + 5 a = 6 and c = 5; Outer + Inner = 17. Factors of 6 Factors of 5 Outer + Inner 1 and 6 1 and 51(5) + 6(1) = 11 2 and 3 1 and 52(5) + 3(1) = 13 3 and 2 1 and 53(5) + 2(1) = 17   (3x + 1)(2x + 5) Check (3x + 1)(2x + 5) = 6x 2 + 15x + 2x + 5 = 6x 2 + 17x + 5 Use the FOIL method. ( x + )( x + )

19. 8-4 Factoring ax 2 + bx + c Check It Out! Example 2b Factor each trinomial. Check your answer. 9x 2 – 15x + 4 a = 9 and c = 4; Outer + Inner = – 15. Factors of 9 Factors of 4 Outer + Inner 3 and 3 –1 and – 4 3(–4) + 3(–1) = –15 3 and 3 – 2 and – 2 3(–2) + 3(–2) = –12 3 and 3 – 4 and – 13(–1) + 3(–4)= –15  (3x – 4)(3x – 1) Check (3x – 4)(3x – 1) = 9x 2 – 3x – 12x + 4 = 9x 2 – 15x + 4 Use the FOIL method. ( x + )( x + )

20. 8-4 Factoring ax 2 + bx + c Factor each trinomial. Check your answer. 3x 2 + 13x + 12 a = 3 and c = 12; Outer + Inner = 13. Factors of 3 Factors of 12 Outer + Inner 1 and 3 1 and 12 1(12) + 3(1) = 15 1 and 3 2 and 61(6) + 3(2) = 12 1 and 3 3 and 41(4) + 3(3) = 13   (x + 3)(3x + 4) Check (x + 3)(3x + 4) = 3x 2 + 4x + 9x + 12 = 3x 2 + 13x + 12 Use the FOIL method. Check It Out! Example 2c ( x + )( x + )

21. 8-4 Factoring ax 2 + bx + c 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.

22. 8-4 Factoring ax 2 + bx + c Additional Example 3A: Factoring ax 2 + bx + c When c is Negative Factor each trinomial. Check your answer. 3n 2 + 11n – 4 ( n + )( n+ ) a = 3 and c = – 4; Outer + Inner = 11. (n + 4)(3n – 1) Check (n + 4)(3n – 1) = 3n 2 – n + 12n – 4 = 3n 2 + 11n – 4 Use the FOIL method. Factors of 3 Factors of –4 Outer + Inner 1 and 3 –1 and 4 1(4) + 3(–1) = 1 1 and 3 –2 and 2 1(2) + 3(–2) = – 4 1 and 3 –4 and 11(1) + 3(–4) = –11  1 and 3 4 and –11(–1) + 3(4) = 11  

23. 8-4 Factoring ax 2 + bx + c Factor each trinomial. Check your answer. 2x 2 + 9x – 18 ( x + )( x+ ) a = 2 and c = –18; Outer + Inner = 9. Factors of 2 Factors of –18 Outer + Inner 1 and 2 18 and –1 1(–1) + 2(18) = 35 1 and 2 9 and –2 1(–2) + 2(9) = 16 1 and 2 6 and –31(–3) + 2(6) = 9   (x + 6)(2x – 3) Check (x + 6)(2x – 3) = 2x 2 – 3x + 12x – 18 = 2x 2 + 9x – 18 Use the FOIL method. Additional Example 3B: Factoring ax 2 + bx + c When c is Negative

24. 8-4 Factoring ax 2 + bx + c Factor each trinomial. Check your answer. 4x 2 – 15x – 4 ( x + )( x+ ) a = 4 and c = –4; Outer + Inner = –15. Factors of 4 Factors of – 4 Outer + Inner 1 and 4 –1 and 4 1(4) – 1(4) = 0 1 and 4 –2 and 2 1(2) – 2(4) = –6 1 and 4 –4 and 1 1(1) – 4(4) = –15   (x – 4)(4x + 1) Use the FOIL method. Check (x – 4)(4x + 1) = 4x 2 + x – 16x – 4 = 4x 2 – 15x – 4 Additional Example 3C: Factoring ax 2 + bx + c When c is Negative

25. 8-4 Factoring ax 2 + bx + c Check It Out! Example 3a Factor each trinomial. Check your answer. 6x 2 + 7x – 3 ( x + )( x+ ) a = 6 and c = –3; Outer + Inner = 7. Factors of 6 Factors of –3 Outer + Inner 6 and 1 1 and –3 6(–3) + 1(1) = –17 6 and 1 3 and –1 6(–1) + 1(3) = – 3    3 and 2 3(–3) + 2(1) = –7 3 and 2 3(–1) + 2(3) = 3  1 and –3 3 and –1 2 and 3 2(–3) + 3(1) = –3 2 and 3 1(–2) + 3(3) = 7  1 and –3 3 and –1 (2x + 3)(3x – 1) Check (2x + 3)(3x – 1) = 6x 2 – 2x + 9x – 3 Use the FOIL method. = 6x + 7x – 3

26. 8-4 Factoring ax 2 + bx + c Check It Out! Example 3b Factor each trinomial. Check your answer. 4n 2 – n – 3 ( n + )( n+ ) a = 4 and c = –3; Outer + Inner = –1. (n – 1) (4n + 3) Use the FOIL method. Factors of 4 Factors of –3 Outer + Inner 1 and 4 1 and –3 1(–3) + 1(4) = 1 1 and 4 –1 and 3 1(3) – 1(4) = –1  Check (n – 1)(4n + 3) = 4n 2 + 3n – 4n – 3 = 4n 2 – n – 3

27. 8-4 Factoring ax 2 + bx + c When the leading coefficient is negative, factor out – 1 from each term before using other factoring methods.

28. 8-4 Factoring ax 2 + bx + c When you factor out –1 in an early step, you must carry it through the rest of the steps and into the answer. Caution!

29. 8-4 Factoring ax 2 + bx + c Additional Example 4: Factoring ax 2 + bx + c When a is Negative Factor –2x 2 – 5x – 3. –1(2x 2 + 5x + 3) – 1( x + )( x+ ) Factor out –1. a = 2 and c = 3; Outer + Inner = 5 Factors of 2 Factors of 3 Outer + Inner 1 and 2 3 and 1 1(1) + 3(2) = 7  1 and 2 1 and 3 1(3) + 1(2) = 5 – 1(x + 1)(2x + 3) (x + 1)(2x + 3)

30. 8-4 Factoring ax 2 + bx + c Check It Out! Example 4a Factor each trinomial. Check your answer. – 6x 2 – 17x – 12 – 1(6x 2 + 17x + 12) – 1( x + )( x+ ) Factor out –1. a = 6 and c = 12; Outer + Inner = 17 Factors of 6 Factors of 12 Outer + Inner 2 and 3 4 and 3 2(3) + 3(4) = 18  2 and 3 3 and 4 2(4) + 3(3) = 17 (2x + 3)(3x + 4) – 1(2x + 3)(3x + 4) Check – 1(2x + 3)(3x + 4) = – 6x 2 – 8x – 9x – 12 = – 6x 2 – 17x – 12

31. 8-4 Factoring ax 2 + bx + c Check It Out! Example 4b Factor each trinomial. Check your answer. – 3x 2 – 17x – 10 – 1(3x 2 + 17x + 10) – 1( x + )( x+ ) Factor out –1. a = 3 and c = 10; Outer + Inner = 17) Factors of 3 Factors of 10 Outer + Inner 1 and 3 2 and 5 1(5) + 3(2) = 11  1 and 3 5 and 2 1(2) + 3(5) = 17 (x + 5)(3x + 2) – 1(x + 5)(3x + 2) Check – 1(x + 5)(3x + 2) = – 3x 2 – 2x – 15x – 10 = – 3x 2 – 17x – 10

32. 8-4 Factoring ax 2 + bx + c Lesson Quiz Factor each trinomial. Check your answer. 1. 5x 2 + 17x + 6 2. 2x 2 + 5x – 12 3. 6x 2 – 23x + 7 4. –4x 2 + 11x + 20 5. –2x 2 + 7x – 3 6. 8x 2 + 27x + 9 (–x + 4)(4x + 5) (3x – 1)(2x – 7) (2x – 3)(x + 4) (5x + 2)(x + 3) ( – 2x + 1)(x – 3) (8x + 3)(x + 3)