7-4 Factoring ax 2 + bx+ c 7-4 Factoring ax 2 +bx+c

Transcription

1 7-4 Factoring ax 2 +bx+c Warm Up Lesson Presentation Lesson Quiz Algebra 1

2 Warm Up Find each product. 1. (x 2)(2x + 7) 2. (3y+ 4)(2y + 9) 3. (3n 5)(n 7) 2x 2 + 3x 14 6y y n 2 26n+ 35 Find each trinomial. 4. x 2 +4x z z h 2 17h+ 72 (x 4)(x+ 8) (z+ 3)(z+ 12) (h 8)(h 9)

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

4 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.

5 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 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 anext to the x s and two numbers that are factors of cin the other blanks. Multiply the binomials to see if you are correct. (3x+2)(2x+5) = 6x x+10

7 Example 1: Factoring ax 2 + bx+ cby Guess and Check Factor 6x x+ 4 by guess and check. ( + )( + ) Write two sets of parentheses. ( x + )( x+ ) The first term is 6x 2, so at least one variable term has a coefficient other than 1. The coefficient of the x 2 term is 6. The constant term in the trinomial is 4. (2x+ 4)(3x + 1) = 6x x+ 4 (1x+ 4)(6x + 1) = 6x x+ 4 (1x+ 2)(6x + 2) = 6x x+ 4 (1x+ 1)(6x + 4) = 6x x+ 4 (3x+ 4)(2x + 1) = 6x x+ 4 Try factors of 6 for the coefficients and factors of 4 for the constant terms.

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

9 Check It Out!Example 1a Factor each trinomial by guess and check. 6x x+ 3 ( + )( + ) Write two sets of parentheses. ( x + )( x+ ) The first term is 6x 2, so at least one variable term has a coefficient other than 1. The coefficient of the x 2 term is 6. The constant term in the trinomial is 3. (2x+ 1)(3x + 3) = 6x 2 + 9x+ 3 (1x+ 3)(6x + 1) = 6x x+ 3 (1x+ 1)(6x + 3) = 6x 2 + 9x+ 3 (3x+ 1)(2x + 3) = 6x x+ 3 Try factors of 6 for the coefficients and factors of 3 for the constant terms.

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

11 Check It Out!Example 1b Factor each trinomial by guess and check. 3x 2 2x 8 ( + )( + ) Write two sets of parentheses. ( x + )( x+ ) The first term is 3x 2, so at least one variable term has a coefficient other than 1. The coefficient of the x 2 term is 3. The constant term in the trinomial is 8. (1x 1)(3x + 8) = 3x 2 + 5x 8 (1x 4)(3x + 2) = 3x 2 10x 8 (1x 8)(3x + 1) = 3x 2 23x 8 (1x 2)(3x + 4) = 3x 2 2x 8 Try factors of 3 for the coefficients and factors of 8 for the constant terms.

12 Check It Out! Example 1b Continued Factor each trinomial by guess and check. 3x 2 2x 8 ( + )( + ) Write two sets of parentheses. ( x + )( x+ ) The first term is 3x 2, so at least one variable term has a coefficient other than 1. The factors of 3x 2 2x 8are (x 2)(3x+ 4). 3x 2 2x 8= (x 2)(3x+ 4)

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

14 Since you need to check all the factors of aand 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. Product = a Product = c ( X+ )( x+ ) =ax 2 +bx+c Sum of outer and inner products = b

15 Example 2A: Factoring ax 2 + bx+ cwhen cis Positive Factor each trinomial. Check your answer. 2x x+ 21 ( x+ )( x+ ) a = 2 and c = 21, Outer + Inner = 17. Factors of 2 Factors of 21 Outer + Inner 1 and 2 1 and 21 1(21) + 2(1) = 23 1 and 2 21 and 1 1(1) + 2(21) = 43 1 and 2 3 and 7 1(7) + 2(3) = 13 1 and 2 7 and 3 1(3) + 2(7) = 17 (x+ 7)(2x+ 3) Use the Foil method. Check (x+ 7)(2x+ 3)= 2x 2 + 3x+ 14x+ 21 = 2x x+ 21

16 Remember! When bis negative and cis positive, the factors of care both negative.

17 Example 2B: Factoring ax 2 + bx+ cwhen cis Positive Factor each trinomial. Check your answer. 3x 2 16x+ 16 ( x+ )( x+ ) a = 3 and c = 16, Outer + Inner = 16. Factors of 3 Factors of 16 Outer + Inner 1 and 3 1 and 16 1( 16) + 3( 1) = 19 1 and 3 2 and 8 1( 8) + 3( 2) = 14 1 and 3 4 and 4 1( 4) + 3( 4)= 16 (x 4)(3x 4) Check (x 4)(3x 4) = 3x 2 4x 12x + 16 = 3x 2 16x + 16 Use the Foil method.

18 Check It Out!Example 2a Factor each trinomial. Check your answer. 6x x+ 5 a = 6 and c = 5, ( x+ )( x+ ) Outer + Inner = 17. Factors of 6 Factors of 5 Outer + Inner 1 and 6 1 and 5 1(5) + 6(1) = 11 2 and 3 1 and 5 2(5) + 3(1) = 13 3 and 2 1 and 5 3(5) + 2(1) = 17 (3x+ 1)(2x+ 5) Use the Foil method. Check(3x+ 1)(2x+ 5)= 6x x+ 2x+ 5 = 6x x+ 5

19 Check It Out!Example 2b Factor each trinomial. Check your answer. 9x 2 15x+ 4 a = 9 and c = 4, ( x+ )( x+ ) 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 1 3( 1) + 3( 4)= 15 (3x 4)(3x 1) Use the Foil method. Check (3x 4)(3x 1) = 9x 2 3x 12x + 4 = 9x 2 15x + 4

20 Check It Out!Example 2c Factor each trinomial. Check your answer. 3x x+ 12 a = 3 and c = 12, ( x+ )( x+ ) 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 6 1(6) + 3(2) = 12 1 and 3 3 and 4 1(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.

21 When cis negative, one factor of cwill 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 Example 3A: Factoring ax 2 + bx+ cwhen cis Negative Factor each trinomial. Check your answer. 3n n 4 ( n+ )( n+ ) Factors of 3 Factors of 4 Outer + Inner a = 3 and c = 4, Outer + Inner = 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 1 1(1) + 3( 4) = 11 1 and 3 4 and 1 1( 1) + 3(4) = 11 (n+ 4)(3n 1) Use the Foil method. Check (n+ 4)(3n 1) = 3n 2 n + 12n 4 = 3n n 4

23 Example 3B: Factoring ax 2 + bx+ cwhen cis Negative 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 3 1( 3) + 2(6) = 9 (x+ 6)(2x 3) Use the Foil method. Check (x+ 6)(2x 3) = 2x 2 3x+ 12x 18 = 2x 2 + 9x 18

24 Example 3C: Factoring ax 2 + bx+ cwhen cis Negative 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) + 4( 1) = 0 1 and 4 2 and 2 1(2) + 4( 2) = 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

25 Check It Out!Example 3a Factor each trinomial. Check your answer. 6x 2 + 7x 3 ( x+ )( x+ ) Factors of 6 Factors of 3 Outer + Inner a = 6 and c = 3, Outer + Inner = 7. 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 1 and 3 3( 3) + 2(1) = 7 3 and 2 3 and 1 3( 1) + 2(3) = 3 2 and 3 1 and 3 2( 3) + 3(1) = 3 2 and 3 3 and 1 2( 1) + 3(3) = 7 (3x 1)(2x+ 3) Use the Foil method. Check (3x 1)(2x+ 3) = 6x 2 + 9x 2x 3 = 6x 2 + 7x 3

26 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. Factors of 4 Factors of 3 Outer + Inner 1 and 4 1 and 3 1( 3) + 4(1) = 1 1 and 4 1 and 3 1(3) 4(1) = 1 (4n + 3)(n 1) Use the Foil method. Check (4n + 3)(n 1) = 4n 2 4n+ 3n 3 = 4n 2 n 3

27 When the leading coefficient is negative, factor out 1 from each term before using other factoring methods.

28 Caution When you factor out 1 in an early step, you must carry it through the rest of the steps.

29 Example 4A: Factoring ax 2 + bx +cwhen ais Negative Factor 2x 2 5x 3. 1(2x 2 + 5x+ 3) 1( x+ )( x+ ) Factors of 2 Factors of 3 Factor out 1. a = 2 and c = 3; Outer + Inner = 5 Outer + Inner 1 and 2 3 and 1 1(1) + 3(2) = 7 1 and 2 1 and 3 1(3) + 1(2) = 5 (x+ 1)(2x + 3) 1(x+ 1)(2x + 3)

30 Factor each trinomial. 6x 2 17x 12 Check It Out!Example 4a 1(6x x+ 12) 1( x+ )( x+ ) Factors of 6 Factors of 12 Factor out 1. a = 6 and c = 12; Outer + Inner = 17 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 Factor each trinomial. 3x 2 17x 10 Check It Out!Example 4b 1(3x x+ 10) 1( x+ )( x+ ) Factors of 3 Factors of 10 Factor out 1. a = 3 and c = 10; Outer + Inner = 17) 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 Lesson Quiz Factor each trinomial. Check your answer. 1.5x x x 2 + 5x x 2 23x+ 7 (5x + 2)(x+ 3) (2x 3)(x + 4) (3x 1)(2x 7) 4. 4x x+ 20 ( x+ 4)(4x+ 5) 5. 2x 2 + 7x 3 6.8x x + 9 ( 2x+ 1)(x 3) (8x+ 3)(x+ 3)