2 2 Factoring axbx bxcc Warm Up Lesson

2 2 Factoring ax++bx bx++cc Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 Holt Mc. Dougal

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

Factoring ax 2 + bx + c Objective Factor quadratic trinomials of the form ax 2 + bx + c. Holt Mc. Dougal Algebra 1

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. Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c When you multiply (3 x + 2)(2 x + 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. (3 x + 2)(2 x + 5) = 6 x 2 + 19 x + 10 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c To factor a trinomial like ax 2 + bx + c into its binomial factors, write two sets of parentheses ( 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. (3 x + 2)(2 x + 5) = 6 x 2 + 19 x + 10 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Example 1: Factoring ax 2 + bx + c by Guess and Check Factor 6 x 2 + 11 x + 4 by guess and check. ( + ) Write two sets of parentheses. 2 ( x + ) The first term is 6 x , 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. (2 x + 4)(3 x + 1) = 6 x 2 + 14 x + 4 Try factors of 6 for the 2 (1 x + 4)(6 x + 1) = 6 x + 25 x + 4 coefficients and (1 x + 2)(6 x + 2) = 6 x 2 + 14 x + 4 factors of 4 for the (1 x + 1)(6 x + 4) = 6 x 2 + 10 x + 4 constant terms. (3 x + 4)(2 x + 1) = 6 x 2 + 11 x + 4 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Example 1 Continued Factor 6 x 2 + 11 x + 4 by guess and check. ( + ) Write two sets of parentheses. 2 ( x + ) The first term is 6 x , so at least one variable term has a coefficient other than 1. The factors of 6 x 2 + 11 x + 4 are (3 x + 4) and (2 x + 1). 6 x 2 + 11 x + 4 = (3 x + 4)(2 x + 1) Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 1 a Factor each trinomial by guess and check. 6 x 2 + 11 x + 3 ( + ) Write two sets of parentheses. 2 ( x + ) The first term is 6 x , 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. (2 x + 1)(3 x + 3) = 6 x 2 + 9 x + 3 Try factors of 6 for the coefficients and (1 x + 3)(6 x + 1) = 6 x 2 + 19 x + 3 factors of 3 for the 2 (1 x + 1)(6 x + 3) = 6 x + 9 x + 3 constant terms. (3 x + 1)(2 x + 3) = 6 x 2 + 11 x + 3 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 1 a Continued Factor each trinomial by guess and check. 6 x 2 + 11 x + 3 ( + ) Write two sets of parentheses. 2 ( x + ) The first term is 6 x , so at least one variable term has a coefficient other than 1. The factors of 6 x 2 + 11 x + 3 are (3 x + 1)(2 x + 3). 6 x 2 + 11 x + 3 = (3 x + 1)(2 x +3) Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 1 b Factor each trinomial by guess and check. 3 x 2 – 2 x – 8 ( + ) Write two sets of parentheses. 2, so at least The first term is 3 x ( x + ) 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. (1 x – 1)(3 x + 8) = 3 x 2 + 5 x – 8 Try factors of 3 for the coefficients and 2 (1 x – 4)(3 x + 2) = 3 x – 10 x – 8 factors of 8 for the (1 x – 8)(3 x + 1) = 3 x 2 – 23 x – 8 constant terms. (1 x – 2)(3 x + 4) = 3 x 2 – 2 x – 8 Holt Mc. Dougal Algebra 1

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

Factoring 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. Product = c Product = a ( X+ )( x+ ) = ax 2 + bx + c Sum of outer and inner products = b Holt Mc. Dougal Algebra 1

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. Product = c Product = a ( X+ )( x+ ) = ax 2 + bx + c Sum of outer and inner products = b Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Example 2 A: Factoring ax 2 + bx + c When c is Positive Factor each trinomial. Check your answer. 2 x 2 + 17 x + 21 ( x+ )( x+ a = 2 and c = 21, ) Outer + Inner = 17. Factors of 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)(2 x + 3) Check (x + 7)(2 x + 3) = 2 x 2 + 3 x + 14 x + 21 = 2 x 2 + 17 x + 21 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Remember! When b is negative and c is positive, the factors of c are both negative. Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Example 2 B: Factoring ax 2 + bx + c When c is Positive Factor each trinomial. Check your answer. 3 x 2 – 16 x + 16 ( 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)(3 x – 4) Use the Foil method. Check (x – 4)(3 x – 4) = 3 x 2 – 4 x – 12 x + 16 = 3 x 2 – 16 x + 16 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 2 a Factor each trinomial. Check your answer. 6 x 2 + 17 x + 5 ( x+ )( x+ Factors of 6 Factors of 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. (3 x + 1)(2 x + 5) Check (3 x + 1)(2 x + 5) = 6 x 2 + 15 x + 2 x + 5 = 6 x 2 + 17 x + 5 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 2 b Factor each trinomial. Check your answer. 9 x 2 – 15 x + 4 ( 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 (3 x – 4)(3 x – 1) Use the Foil method. Check (3 x – 4)(3 x – 1) = 9 x 2 – 3 x – 12 x + 4 = 9 x 2 – 15 x + 4 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 2 c Factor each trinomial. Check your answer. 3 x 2 + 13 x + 12 ( 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 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. (x + 3)(3 x + 4) Check (x + 3)(3 x + 4) = 3 x 2 + 4 x + 9 x + 12 = 3 x 2 + 13 x + 12 Holt Mc. Dougal Algebra 1

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. Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Example 3 A: Factoring ax 2 + bx + c When c is Negative Factor each trinomial. Check your answer. 3 n 2 + 11 n – 4 ( n+ )( n+ Factors of 3 Factors of – 4 1 and 3 – 1 and 4 1 and 3 – 2 and 2 – 4 and 1 1 and 3 4 and – 1 1 and 3 ) a = 3 and c = – 4, Outer + Inner = 11. Outer + Inner 1(4) + 3(– 1) = 1 1(2) + 3(– 2) = – 4 1(1) + 3(– 4) = – 11 1(– 1) + 3(4) = 11 (n + 4)(3 n – 1) Use the Foil method. Check (n + 4)(3 n – 1) = 3 n 2 – n + 12 n – 4 = 3 n 2 + 11 n – 4 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Example 3 B: Factoring ax 2 + bx + c When c is Negative Factor each trinomial. Check your answer. 2 x 2 + 9 x – 18 ( 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)(2 x – 3) Use the Foil method. Check (x + 6)(2 x – 3) = 2 x 2 – 3 x + 12 x – 18 = 2 x 2 + 9 x – 18 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Example 3 C: Factoring ax 2 + bx + c When c is Negative Factor each trinomial. Check your answer. 4 x 2 – 15 x – 4 ( x+ ) Factors of 4 Factors of – 4 1 and 4 – 2 and 2 – 4 and 1 1 and 4 (x – 4)(4 x + 1) a = 4 and c = – 4, Outer + Inner = – 15. Outer + Inner 1(4) + 4(– 1) = 0 1(2) + 4(– 2) = – 6 1(1) + 4(– 4) = – 15 Use the Foil method. Check (x – 4)(4 x + 1) = 4 x 2 + x – 16 x – 4 = 4 x 2 – 15 x – 4 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 3 a Factor each trinomial. Check your answer. 6 x 2 + 7 x – 3 a = 6 and c = – 3, ( x + )( x+ ) 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 2(– 1) + 3(3) = 7 2 and 3 3 and – 1 Use the Foil method. (3 x – 1)(2 x + 3) Check (3 x – 1)(2 x + 3) = 6 x 2 + 9 x – 2 x – 3 = 6 x 2 + 7 x – 3 Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 3 b Factor each trinomial. Check your answer. 4 n 2 – n – 3 ( n+ ) a = 4 and c = – 3, Outer + Inner = – 1. Factors of 4 Factors of – 3 Outer + Inner 1 and 4 1(– 3) + 4(1) = 1 1 and – 3 1(3) – 4(1) = – 1 1 and 4 – 1 and 3 (4 n + 3)(n – 1) Use the Foil method. Check (4 n + 3)(n – 1) = 4 n 2 – 4 n + 3 n – 3 = 4 n 2 – n – 3 Holt Mc. Dougal Algebra 1

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

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

Factoring ax 2 + bx + c Example 4 A: Factoring ax 2 + bx + c When a is Negative Factor – 2 x 2 – 5 x – 3. – 1(2 x 2 + 5 x + 3) – 1( x+ ) Factors of 2 Factors of 3 Factor out – 1. a = 2 and c = 3; Outer + Inner = 5 Outer + Inner 5 1 and 2 3 and 1 1(1) + 3(2) = 7 1 and 2 1 and 3 1(3) + 1(2) = (x + 1)(2 x + 3) – 1(x + 1)(2 x + 3) Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 4 a Factor each trinomial. – 6 x 2 – 17 x – 12 – 1(6 x 2 + 17 x + 12) – 1( x+ ) Factors of 6 Factors of 12 Factor out – 1. a = 6 and c = 12; Outer + Inner = 17 Outer + Inner 17 2 and 3 4 and 3 2(3) + 3(4) = 18 2 and 3 3 and 4 2(4) + 3(3) = (2 x + 3)(3 x + 4) – 1(2 x + 3)(3 x + 4) Holt Mc. Dougal Algebra 1

Factoring ax 2 + bx + c Check It Out! Example 4 b Factor each trinomial. – 3 x 2 – 17 x – 10 – 1(3 x 2 + 17 x + 10) – 1( x+ ) Factors of 3 Factors of 10 Factor out – 1. a = 3 and c = 10; Outer + Inner = 17) Outer + Inner 17 1 and 3 2 and 5 1(5) + 3(2) = 11 1 and 3 5 and 2 1(2) + 3(5) = (3 x + 2)(x + 5) – 1(3 x + 2)(x + 5) Holt Mc. Dougal Algebra 1

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