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2 2 8-4 ++bx 8-4 Factoring Factoringax ax bx++cc Warm Up Lesson Presentation Lesson Quiz Holt Algebra Holt Algebra 11 8-4 Factoring ax2 + bx + c Warm Up Find each product. 1. (x – 2)(2x + 7) 2x2 + 3x – 14 2. (3y + 4)(2y + 9) 6y2 + 35y + 36 3. (3n – 5)(n – 7) 3n2 – 26n + 35 Find each trinomial. 4. x2 +4x – 32 (x – 4)(x + 8) 5. z2 + 15z + 36 (z + 3)(z + 12) 6. h2 – 17h + 72 (h – 8)(h – 9) Holt Algebra 1 8-4 Factoring ax2 + bx + c Objective Factor quadratic trinomials of the form ax2 + bx + c. Holt Algebra 1 8-4 Factoring ax2 + bx + c In the previous lesson you factored trinomials of the form x2 + bx + c. Now you will factor trinomials of the form ax + bx + c, where a ≠ 0. Holt Algebra 1 8-4 Factoring ax2 + bx + c When you multiply (3x + 2)(2x + 5), the coefficient of the x2-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) = 6x2 + 19x + 10 Holt Algebra 1 8-4 Factoring ax2 + bx + c To factor a trinomial like ax2 + 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) = 6x2 + 19x + 10 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 1: Factoring ax2 + bx + c by Guess and Check Factor 6x2 + 11x + 4 by guess and check. ( + )( + ) Write two sets of parentheses. 2 ( x + )( x + ) The first term is 6x , so at least one variable term has a coefficient other than 1. The coefficient of the x2 term is 6. The constant term in the trinomial is 4. (2x + 4)(3x + 1) = 6x2 + 14x + 4 Try factors of 6 for the 2 (1x + 4)(6x + 1) = 6x + 25x + 4 coefficients and (1x + 2)(6x + 2) = 6x2 + 14x + 4 factors of 4 for the (1x + 1)(6x + 4) = 6x2 + 10x + 4 constant terms. (3x + 4)(2x + 1) = 6x2 + 11x + 4 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 1 Continued Factor 6x2 + 11x + 4 by guess and check. ( + )( + ) ( x + )( x + ) Write two sets of parentheses. The first term is 6x2, so at least one variable terms has a coefficient other than 1. The factors of 6x2 + 11x + 4 are (3x + 4) and (2x + 1). 6x2 + 11x + 4 = (3x + 4)(2x + 1) Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 1a Factor each trinomial by guess and check. 6x2 + 11x + 3 ( + )( + ) Write two sets of parentheses. ( x + )( x + ) The first term is 6x2, so at least one variable term has a coefficient other than 1. The coefficient of the x2 term is 6. The constant term in the trinomial is 3. (2x + 1)(3x + 3) = 6x2 + 9x + 3 Try factors of 6 for the coefficients and (1x + 3)(6x + 1) = 6x2 + 19x + 3 factors of 3 for the 2 (1x + 1)(6x + 3) = 6x + 9x + 3 constant terms. (3x + 1)(2x + 3) = 6x2 + 11x + 3 Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 1a Continued Factor each trinomial by guess and check. 6x2 + 11x + 3 ( + )( + ) ( x + )( x + ) Write two sets of parentheses. The first term is 6x2, so at least one variable term has a coefficient other than 1. The factors of 6x2 + 11x + 3 are (3x + 1)(2x + 3). 6x2 + 11x + 3 = (3x + 1)(2x +3) Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 1b Factor each trinomial by guess and check. 3x2 – 2x – 8 ( + )( + ) Write two sets of parentheses. 2, so at least The first term is 3x ( x + )( x + ) one variable term has a coefficient other than 1. The coefficient of the x2 term is 3. The constant term in the trinomial is –8. (1x – 1)(3x + 8) = 3x2 + 5x – 8 Try factors of 3 for the coefficients and 2 (1x – 4)(3x + 2) = 3x – 10x – 8 factors of 8 for the (1x – 8)(3x + 1) = 3x2 – 23x – 8 constant terms. (1x – 2)(3x + 4) = 3x2 – 2x – 8 Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 1b Factor each trinomial by guess and check. 3x2 – 2x – 8 ( + )( + ) ( x + )( x + ) Write two sets of parentheses. The first term is 3x2, so at least one variable term has a coefficient other than 1. The factors of 3x2 – 2x – 8 are (x – 2)(3x + 4). 3x2 – 2x – 8 = (x – 2)(3x + 4) Holt Algebra 1 8-4 Factoring ax2 + bx + c So, to factor a2 + 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. Product = c Product = a ( X+ )( x+ ) = ax2 + bx + c Sum of outer and inner products = b Holt Algebra 1 8-4 Factoring ax2 + 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. Product = c Product = a ( X+ )( x+ ) = ax2 + bx + c Sum of outer and inner products = b Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c When c is Positive Factor each trinomial. Check your answer. 2x2 + 17x + 21 ( x+ )( x+ a = 2 and c = 21, ) Outer + Inner = 17. Factors of 2 Factors of 21 1 and 21 1 and 2 21 and 1 1 and 2 3 and 7 1 and 2 7 and 3 1 and 2 Outer + Inner 1(21) + 2(1) = 23 1(1) + 2(21) = 43 1(7) + 2(3) = 13 1(3) + 2(7) = 17 Use the Foil method. (x + 7)(2x + 3) Check (x + 7)(2x + 3) = 2x2 + 3x + 14x + 21 = 2x2 + 17x + 21 Holt Algebra 1 8-4 Factoring ax2 + bx + c Remember! When b is negative and c is positive, the factors of c are both negative. Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2B: Factoring ax2 + bx + c When c is Positive Factor each trinomial. Check your answer. 3x2 – 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 – 4 and – 4 1( – 4) + 3(– 4)= –16 1 and 3 (x – 4)(3x – 4) Use the Foil method. Check (x – 4)(3x – 4) = 3x2 – 4x – 12x + 16 = 3x2 – 16x + 16 Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 2a Factor each trinomial. Check your answer. 6x2 + 17x + 5 ( x+ )( x+ Factors of 6 Factors of 5 1 and 5 1 and 6 1 and 5 2 and 3 1 and 5 3 and 2 ) a = 6 and c = 5, Outer + Inner = 17. Outer + Inner 1(5) + 6(1) = 11 2(5) + 3(1) = 13 3(5) + 2(1) = 17 Use the Foil method. (3x + 1)(2x + 5) Check (3x + 1)(2x + 5) = 6x2 + 15x + 2x + 5 = 6x2 + 17x + 5 Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 2b Factor each trinomial. Check your answer. 9x2 – 15x + 4 ( x + )( x + ) 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 – 4 and – 1 3(–1) + 3(– 4)= –15 3 and 3 (3x – 4)(3x – 1) Use the Foil method. Check (3x – 4)(3x – 1) = 9x2 – 3x – 12x + 4 = 9x2 – 15x + 4 Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 2c Factor each trinomial. Check your answer. 3x2 + 13x + 12 ( x+ )( x+ ) Factors of 3 Factors of 12 1 and 3 1 and 12 2 and 6 1 and 3 3 and 4 1 and 3 (x + 3)(3x + 4) a = 3 and c = 12, Outer + Inner = 13. Outer + Inner 1(12) + 3(1) = 15 1(6) + 3(2) = 12 1(4) + 3(3) = 13 Use the Foil method. Check (x + 3)(3x + 4) = 3x2 + 4x + 9x + 12 = 3x2 + 13x + 12 Holt Algebra 1 8-4 Factoring ax2 + 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. Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 3A: Factoring ax2 + bx + c When c is Negative Factor each trinomial. Check your answer. 3n2 + 11n – 4 ( n+ )( n+ ) a = 3 and c = – 4, Outer + Inner = 11 . Factors of 3 Factors of 4 Outer + Inner 1 and 3 –1 and 4 1(4) + 3(–1) = 1 1(2) + 3(–2) = – 4 1 and 3 –2 and 2 –4 and 1 1(1) + 3(–4) = –11 1 and 3 4 and –1 1(–1) + 3(4) = 11 1 and 3 (n + 4)(3n – 1) Use the Foil method. Check (n + 4)(3n – 1) = 3n2 – n + 12n – 4 = 3n2 + 11n – 4 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 3B: Factoring ax2 + bx + c When c is Negative Factor each trinomial. Check your answer. 2x2 + 9x – 18 ( x+ )( x+ ) a = 2 and c = –18, Outer + Inner = 9. Factors of 2 Factors of – 18 Outer + Inner 1(– 1) + 2(18) = 35 1 and 2 18 and –1 1(– 2) + 2(9) = 16 1 and 2 9 and –2 6 and –3 1(– 3) + 2(6) = 9 1 and 2 (x + 6)(2x – 3) Use the Foil method. Check (x + 6)(2x – 3) = 2x2 – 3x + 12x – 18 = 2x2 + 9x – 18 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 3C: Factoring ax2 + bx + c When c is Negative Factor each trinomial. Check your answer. 4x2 – 15x – 4 ( x+ )( x+ Factors of 4 Factors of – 4 1 and 4 –1 and 4 1 and 4 –2 and 2 –4 and 1 1 and 4 (x – 4)(4x + 1) ) a = 4 and c = –4, Outer + Inner = –15. Outer + Inner 1(4) – 1(4) = 0 1(2) – 2(4) = –6 1(1) – 4(4) = –15 Use the Foil method. Check (x – 4)(4x + 1) = 4x2 + x – 16x – 4 = 4x2 – 15x – 4 Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 3a Factor each trinomial. Check your answer. 6x2 + 7x – 3 ( x+ )( x+ ) a = 6 and c = –3, Outer + Inner = 7. Factors of 6 Factors of – 3 Outer + Inner 6(–3) + 1(1) = –17 6 and 1 1 and –3 6(–1) + 1(3) = – 3 6 and 1 3 and –1 3(–3) + 2(1) = – 7 3 and 2 1 and –3 3(–1) + 2(3) = 3 3 and 2 3 and –1 2(–3) + 3(1) = – 3 2 and 3 1 and –3 1(–2) + 3(3) = 7 2 and 3 3 and –1 (3x – 1)(2x + 3) Use the Foil method. Check (3x – 1)(2x + 3) = 6x2 + 9x – 2x – 3 = 6x + 7x – 3 Holt Algebra 1 8-4 Factoring ax2 + bx + c Check It Out! Example 3b Factor each trinomial. Check your answer. 4n2 – n – 3 ( n+ )( n+ ) a = 4 and c = –3, Outer + Inner = –1. Factors of 4 Factors of –3 Outer + Inner 1 and 4 –1(3) + 1(4) = 1 1 and –3 1(3) –1(4) = – 1 1 and 4 –1 and 3 (4n + 3)(n – 1) Use the Foil method. Check (4n + 3)(n – 1) = 4n2 – 4n + 3n – 3 = 4n2 – n – 3 Holt Algebra 1 8-4 Factoring ax2 + bx + c When the leading coefficient is negative, factor out –1 from each term before using other factoring methods. Holt Algebra 1 8-4 Factoring ax2 + bx + c Caution When you factor out –1 in an early step, you must carry it through the rest of the steps. Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 4A: Factoring ax2 + bx + c When a is Negative Factor –2x2 – 5x – 3. –1(2x2 + 5x + 3) –1( x+ )( x+ ) 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 (x + 1)(2x + 3) –1(x + 1)(2x + 3) Holt Algebra 1 Factor out –1. a = 2 and c = 3; Outer + Inner = 5 8-4 Factoring ax2 + bx + c Check It Out! Example 4a Factor each trinomial. Check your answer. –6x2 – 17x – 12 –1(6x2 + 17x + 12) –1( x+ )( x+ ) Factors of 6 Factors of 12 Outer + Inner 17 2 and 3 4 and 3 2(3) + 3(4) = 18 2 and 3 3 and 4 2(4) + 3(3) = (2x + 3)(3x + 4) –1(2x + 3)(3x + 4) Holt Algebra 1 Factor out –1. a = 6 and c = 12; Outer + Inner = 17 8-4 Factoring ax2 + bx + c Check It Out! Example 4b Factor each trinomial. Check your answer. –3x2 – 17x – 10 –1(3x2 + 17x + 10) –1( x+ )( x+ ) Factors of 3 Factors of 10 Outer + Inner 17 1 and 3 2 and 5 1(5) + 3(2) = 11 1 and 3 5 and 2 1(2) + 3(5) = (3x + 2)(x + 5) –1(3x + 2)(x + 5) Holt Algebra 1 Factor out –1. a = 3 and c = 10; Outer + Inner = 17) 8-4 Factoring ax2 + bx + c Lesson Quiz Factor each trinomial. Check your answer. 1. 5x2 + 17x + 6 (5x + 2)(x + 3) 2. 2x2 + 5x – 12 (2x– 3)(x + 4) 3. 6x2 – 23x + 7 (3x – 1)(2x – 7) 4. –4x2 + 11x + 20 (–x + 4)(4x + 5) 5. –2x2 + 7x – 3 (–2x + 1)(x – 3) 6. 8x2 + 27x + 9 (8x + 3)(x + 3) Holt Algebra 1