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

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 – 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)

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

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.

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 x + 10

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.

8-4 Factoring ax 2 + bx + c Additional Example 1: Factoring ax 2 + bx + c Factor 6x x + 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 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

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

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 x + 3 (1x + 3)(6x + 1) = 6x x + 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 x + 3

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

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

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.

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

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

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 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 FOILmethod.

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

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

8-4 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 + )

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 + )

8-4 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 + )

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.

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 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  

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

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

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

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

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

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!

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)

8-4 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) Check – 1(2x + 3)(3x + 4) = – 6x 2 – 8x – 9x – 12 = – 6x 2 – 17x – 12

8-4 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 (x + 5)(3x + 2) – 1(x + 5)(3x + 2) Check – 1(x + 5)(3x + 2) = – 3x 2 – 2x – 15x – 10 = – 3x 2 – 17x – 10

8-4 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)