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Holt Algebra 1 8-4 Factoring ax 2 + bx + c 8-4 Factoring ax 2 + bx + c Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz.

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Presentation on theme: "Holt Algebra 1 8-4 Factoring ax 2 + bx + c 8-4 Factoring ax 2 + bx + c Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz."— Presentation transcript:

1 Holt Algebra Factoring ax 2 + bx + c 8-4 Factoring ax 2 + bx + c Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2 Holt Algebra 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 – z z h 2 – 17h y y x 2 + 3x – 14 3n 2 – 26n + 35 (z + 3)(z + 12) (x – 4)(x + 8) (h – 8)(h – 9)

3 Holt Algebra Factoring ax 2 + bx + c Factor quadratic trinomials of the form ax 2 + bx + c. Objective

4 Holt Algebra 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 + bx + c, where a 0.

5 Holt Algebra 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 x + 10

6 Holt Algebra Factoring ax 2 + bx + c 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 x + 10

7 Holt Algebra Factoring ax 2 + bx + c Example 1: Factoring ax 2 + bx + c by Guess and Check Factor 6x x + 4 by guess and check. ( + )( + ) Write two sets of parentheses. 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 factors of 6 for the coefficients and factors of 4 for the constant terms. (1x + 4)(6x + 1) = 6x x + 4 (1x + 2)(6x + 2) = 6x x + 4 (1x + 1)(6x + 4) = 6x x + 4 (2x + 4)(3x + 1) = 6x x + 4 (3x + 4)(2x + 1) = 6x x + 4

8 Holt Algebra Factoring ax 2 + bx + c Example 1 Continued Factor 6x x + 4 by guess and check. ( + )( + ) Write two sets of parentheses. The first term is 6x 2, so at least one variable terms has a coefficient other than 1. ( x + )( x + ) 6x x + 4 = (3x + 4)(2x + 1) The factors of 6x x + 4 are (3x + 4) and (2x + 1).

9 Holt Algebra Factoring ax 2 + bx + c Check It Out! Example 1a Factor each trinomial by guess and check. ( + )( + ) Write two sets of parentheses. 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 x + 3 (1x + 3)(6x + 1) = 6x x + 3 (1x + 1)(6x + 3) = 6x 2 + 9x + 3 Try factors of 6 for the coefficients and factors of 3 for the constant terms. (2x + 1)(3x + 3) = 6x 2 + 9x + 3 (3x + 1)(2x + 3) = 6x x + 3

10 Holt Algebra Factoring ax 2 + bx + c Check It Out! Example 1a Continued Factor each trinomial by guess and check. ( + )( + ) Write two sets of parentheses. The first term is 6x 2, so at least one variable term has a coefficient other than 1. ( x + )( x + ) 6x x + 3 The factors of 6x x + 3 are (3x + 1)(2x + 3). 6x x + 3 = (3x + 1)(2x +3)

11 Holt Algebra Factoring ax 2 + bx + c Check It Out! Example 1b Factor each trinomial by guess and check. ( + )( + ) Write two sets of parentheses. 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 factors of 3 for the coefficients and 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 Holt Algebra Factoring ax 2 + bx + c Check It Out! Example 1b Factor each trinomial by guess and check. ( + )( + ) Write two sets of parentheses. 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 factors of 3x 2 – 2x – 8 are (x – 2)(3x + 4). 3x 2 – 2x – 8 = (x – 2)(3x + 4)

13 Holt Algebra Factoring ax 2 + bx + c ( X + )( x + ) = ax 2 + bx + c 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. Sum of outer and inner products = b Product = c Product = a

14 Holt Algebra Factoring ax 2 + bx + c Since you need to check all the factors of a and 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 Holt Algebra Factoring ax 2 + bx + c Example 2A: Factoring ax 2 + bx + c When c is Positive Factor each trinomial. Check your answer. 2x x + 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 x + 21 Use the Foil method.

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

17 Holt Algebra 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 + ) Example 2B: Factoring ax 2 + bx + c When c is Positive

18 Holt Algebra Factoring ax 2 + bx + c Check It Out! Example 2a Factor each trinomial. Check your answer. 6x x + 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 x + 2x + 5 = 6x x + 5 Use the Foil method. ( x + )( x + )

19 Holt Algebra 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 Holt Algebra Factoring ax 2 + bx + c Factor each trinomial. Check your answer. 3x x + 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 x + 12 Use the Foil method. Check It Out! Example 2c ( x + )( x + )

21 Holt Algebra 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 Holt Algebra Factoring ax 2 + bx + c Example 3A: Factoring ax 2 + bx + c When c is Negative Factor each trinomial. Check your answer. 3n n – 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 n – 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 Holt Algebra 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. Example 3B: Factoring ax 2 + bx + c When c is Negative

24 Holt Algebra 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 Example 3C: Factoring ax 2 + bx + c When c is Negative

25 Holt Algebra 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 (3x – 1)(2x + 3) Check (3x – 1)(2x + 3) = 6x 2 + 9x – 2x – 3 Use the Foil method. = 6x + 7x – 3

26 Holt Algebra 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. (4n + 3)(n – 1) 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 (4n + 3)(n – 1) = 4n 2 – 4n + 3n – 3 = 4n 2 – n – 3

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

28 Holt Algebra Factoring ax 2 + bx + c When you factor out –1 in an early step, you must carry it through the rest of the steps. Caution

29 Holt Algebra Factoring ax 2 + bx + c Example 4A: 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 Holt Algebra Factoring ax 2 + bx + c Check It Out! Example 4a Factor each trinomial. Check your answer. – 6x 2 – 17x – 12 – 1(6x x + 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)

31 Holt Algebra Factoring ax 2 + bx + c Check It Out! Example 4b Factor each trinomial. Check your answer. – 3x 2 – 17x – 10 – 1(3x x + 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 (3x + 2)(x + 5) – 1(3x + 2)(x + 5)

32 Holt Algebra Factoring ax 2 + bx + c Lesson Quiz Factor each trinomial. Check your answer. 1. 5x x x 2 + 5x – x 2 – 23x –4x x –2x 2 + 7x – x x + 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)


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