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**8-4 Factoring ax2 + bx + c Warm Up Lesson Presentation Lesson Quiz**

Holt Algebra 1

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Warm Up Find each product. 1. (x – 2)(2x + 7) 2. (3y + 4)(2y + 9) 3. (3n – 5)(n – 7) Find each trinomial. 4. x2 +4x – 32 5. z2 + 15z + 36 6. h2 – 17h + 72 2x2 + 3x – 14 6y2 + 35y + 36 3n2 – 26n + 35 (x – 4)(x + 8) (z + 3)(z + 12) (h – 8)(h – 9)

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Objective Factor quadratic trinomials of the form ax2 + bx + c.

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**In the previous lesson you factored trinomials of the form x2 + 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.

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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

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**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

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**Example 1: Factoring ax2 + bx + c by Guess and Check**

Factor 6x2 + 11x + 4 by guess and check. ( )( ) 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 4. (2x + 4)(3x + 1) = 6x2 + 14x + 4 (1x + 4)(6x + 1) = 6x2 + 25x + 4 Try factors of 6 for the coefficients and factors of 4 for the constant terms. (1x + 2)(6x + 2) = 6x2 + 14x + 4 (1x + 1)(6x + 4) = 6x2 + 10x + 4 (3x + 4)(2x + 1) = 6x2 + 11x + 4

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**Factor 6x2 + 11x + 4 by guess and check.**

Example 1 Continued Factor 6x2 + 11x + 4 by guess and check. ( )( ) Write two sets of parentheses. ( x + )( x + ) 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)

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** 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 factors of 3 for the constant terms. (1x + 3)(6x + 1) = 6x2 + 19x + 3 (1x + 1)(6x + 3) = 6x2 + 9x + 3 (3x + 1)(2x + 3) = 6x2 + 11x + 3

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**Check It Out! Example 1a Continued**

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 factors of 6x2 + 11x + 3 are (3x + 1)(2x + 3). 6x2 + 11x + 3 = (3x + 1)(2x +3)

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** Check It Out! Example 1b**

Factor each trinomial by guess and check. 3x2 – 2x – 8 ( )( ) Write two sets of parentheses. The first term is 3x2, so at least one variable term has a coefficient other than 1. ( x + )( x + ) 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 factors of 8 for the constant terms. (1x – 4)(3x + 2) = 3x2 – 10x – 8 (1x – 8)(3x + 1) = 3x2 – 23x – 8 (1x – 2)(3x + 4) = 3x2 – 2x – 8

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Check It Out! Example 1b Factor each trinomial by guess and check. 3x2 – 2x – 8 ( )( ) Write two sets of parentheses. ( x + )( x + ) 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)

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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 = a Product = c Sum of outer and inner products = b ( X + )( x + ) = ax2 + bx + c

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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 = a Product = c Sum of outer and inner products = b ( X + )( x + ) = ax2 + bx + c

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**Example 2A: Factoring ax2 + bx + c When c is Positive**

Factor each trinomial. Check your answer. 2x2 + 17x + 21 a = 2 and c = 21, Outer + Inner = 17. ( x + )( x + ) Factors of 2 Factors of 21 Outer + Inner 1 and 2 1 and 21 1(21) + 2(1) = 23 21 and 1 1(1) + 2(21) = 43 3 and 7 1(7) + 2(3) = 13 7 and 3 1(3) + 2(7) = 17 (x + 7)(2x + 3) Use the Foil method. Check (x + 7)(2x + 3) = 2x2 + 3x + 14x + 21 = 2x2 + 17x + 21

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**When b is negative and c is positive, the factors of c are both negative.**

Remember!

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**Example 2B: Factoring ax2 + bx + c When c is Positive**

Factor each trinomial. Check your answer. 3x2 – 16x + 16 a = 3 and c = 16, Outer + Inner = –16. ( x + )( x + ) Factors of 3 Factors of 16 Outer + Inner 1 and 3 –1 and –16 1(–16) + 3(–1) = –19 – 2 and – 8 1( – 8) + 3(–2) = –14 – 4 and – 4 1( – 4) + 3(– 4)= –16 (x – 4)(3x – 4) Use the Foil method. Check (x – 4)(3x – 4) = 3x2 – 4x – 12x + 16 = 3x2 – 16x + 16

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** Check It Out! Example 2a**

Factor each trinomial. Check your answer. 6x2 + 17x + 5 a = 6 and c = 5, Outer + Inner = 17. ( x + )( x + ) Factors of 6 Factors of 5 Outer + Inner 1 and 6 1 and 5 1(5) + 6(1) = 11 2 and 3 2(5) + 3(1) = 13 3 and 2 3(5) + 2(1) = 17 (3x + 1)(2x + 5) Use the Foil method. Check (3x + 1)(2x + 5) = 6x2 + 15x + 2x + 5 = 6x2 + 17x + 5

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** Check It Out! Example 2b**

Factor each trinomial. Check your answer. 9x2 – 15x + 4 a = 9 and c = 4, Outer + Inner = –15. ( x + )( x + ) Factors of 9 Factors of 4 Outer + Inner 3 and 3 –1 and – 4 3(–4) + 3(–1) = –15 – 2 and – 2 3(–2) + 3(–2) = –12 – 4 and – 1 3(–1) + 3(– 4)= –15 (3x – 4)(3x – 1) Use the Foil method. Check (3x – 4)(3x – 1) = 9x2 – 3x – 12x + 4 = 9x2 – 15x + 4

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** Check It Out! Example 2c**

Factor each trinomial. Check your answer. 3x2 + 13x + 12 a = 3 and c = 12, Outer + Inner = 13. ( x + )( x + ) Factors of 3 Factors of 12 Outer + Inner 1 and 3 1 and 12 1(12) + 3(1) = 15 2 and 6 1(6) + 3(2) = 12 3 and 4 1(4) + 3(3) = 13 (x + 3)(3x + 4) Use the Foil method. Check (x + 3)(3x + 4) = 3x2 + 4x + 9x + 12 = 3x2 + 13x + 12

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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.

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**Example 3A: Factoring ax2 + bx + c When c is Negative**

Factor each trinomial. Check your answer. 3n2 + 11n – 4 a = 3 and c = – 4, Outer + Inner = 11 . ( n + )( n+ ) Factors of 3 Factors of 4 Outer + Inner 1 and 3 –1 and 4 1(4) + 3(–1) = 1 –2 and 2 1(2) + 3(–2) = – 4 –4 and 1 1(1) + 3(–4) = –11 4 and –1 1(–1) + 3(4) = 11 (n + 4)(3n – 1) Use the Foil method. Check (n + 4)(3n – 1) = 3n2 – n + 12n – 4 = 3n2 + 11n – 4

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**Example 3B: Factoring ax2 + bx + c When c is Negative**

Factor each trinomial. Check your answer. 2x2 + 9x – 18 a = 2 and c = –18, Outer + Inner = 9. ( x + )( x+ ) Factors of 2 Factors of – 18 Outer + Inner 1 and 2 18 and –1 1(– 1) + 2(18) = 35 9 and –2 1(– 2) + 2(9) = 16 6 and –3 1(– 3) + 2(6) = 9 (x + 6)(2x – 3) Use the Foil method. Check (x + 6)(2x – 3) = 2x2 – 3x + 12x – 18 = 2x2 + 9x – 18

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**Example 3C: Factoring ax2 + bx + c When c is Negative**

Factor each trinomial. Check your answer. 4x2 – 15x – 4 a = 4 and c = –4, Outer + Inner = –15. ( x + )( x+ ) Factors of 4 Factors of – 4 Outer + Inner 1 and 4 –1 and 4 1(4) – 1(4) = 0 –2 and 2 1(2) – 2(4) = –6 –4 and 1 1(1) – 4(4) = –15 (x – 4)(4x + 1) Use the Foil method. Check (x – 4)(4x + 1) = 4x2 + x – 16x – 4 = 4x2 – 15x – 4

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** Check It Out! Example 3a Factor each trinomial. Check your answer.**

6x2 + 7x – 3 a = 6 and c = –3, Outer + Inner = 7. ( x + )( x+ ) Factors of 6 Factors of – 3 Outer + Inner 6 and 1 1 and –3 6(–3) + 1(1) = –17 3 and –1 6(–1) + 1(3) = – 3 3 and 2 3(–3) + 2(1) = – 7 3(–1) + 2(3) = 3 2 and 3 2(–3) + 3(1) = – 3 1(–2) + 3(3) = 7 (3x – 1)(2x + 3) Use the Foil method. Check (3x – 1)(2x + 3) = 6x2 + 9x – 2x – 3 = 6x + 7x – 3

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** Check It Out! Example 3b Factor each trinomial. Check your answer.**

4n2 – n – 3 a = 4 and c = –3, Outer + Inner = –1. ( n + )( n+ ) Factors of 4 Factors of –3 Outer + Inner 1 and 4 1 and –3 –1(3) + 1(4) = 1 –1 and 3 1(3) –1(4) = – 1 (4n + 3)(n – 1) Use the Foil method. Check (4n + 3)(n – 1) = 4n2 – 4n + 3n – 3 = 4n2 – n – 3

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

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

Caution

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**Example 4A: Factoring ax2 + bx + c When a is Negative**

Factor –2x2 – 5x – 3. –1(2x2 + 5x + 3) Factor out –1. a = 2 and c = 3; Outer + Inner = 5 –1( x + )( x+ ) Factors of 2 Factors of Outer + Inner 1 and 2 3 and 1 1(1) + 3(2) = 7 1 and 3 1(3) + 1(2) = 5 (x + 1)(2x + 3) –1(x + 1)(2x + 3)

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** Check It Out! Example 4a Factor each trinomial. Check your answer.**

–6x2 – 17x – 12 Factor out –1. –1(6x2 + 17x + 12) a = 6 and c = 12; Outer + Inner = 17 –1( x + )( x+ ) Factors of 6 Factors of Outer + Inner 2 and 3 4 and 3 2(3) + 3(4) = 18 3 and 4 2(4) + 3(3) = 17 (2x + 3)(3x + 4) –1(2x + 3)(3x + 4)

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** Check It Out! Example 4b Factor each trinomial. Check your answer.**

–3x2 – 17x – 10 Factor out –1. –1(3x2 + 17x + 10) a = 3 and c = 10; Outer + Inner = 17) –1( x + )( x+ ) Factors of 3 Factors of Outer + Inner 1 and 3 2 and 5 1(5) + 3(2) = 11 5 and 2 1(2) + 3(5) = 17 (3x + 2)(x + 5) –1(3x + 2)(x + 5)

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Lesson Quiz Factor each trinomial. Check your answer. 1. 5x2 + 17x + 6 2. 2x2 + 5x – 12 3. 6x2 – 23x + 7 4. –4x2 + 11x + 20 5. –2x2 + 7x – 3 6. 8x2 + 27x + 9 (5x + 2)(x + 3) (2x– 3)(x + 4) (3x – 1)(2x – 7) (–x + 4)(4x + 5) (–2x + 1)(x – 3) (8x + 3)(x + 3)

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