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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 ax2 + 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, determine the signs and write two sets of parentheses ( x + )( x + ). Write numbers that are factors of a and numbers that are factors of c. Determine which pairings multiply to give you the correct trinomial. (3x + 2)(2x + 5) = 6x2 + 19x + 10

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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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**Example 1A: Factoring ax2 + bx + c**

Factor 6x2 + 11x + 4. 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 The factors of 6x2 + 11x + 4 are (3x + 4) and (2x + 1).

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** Example 1B Factor each trinomial. 6x2 + 11x + 3**

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

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** Example 1C Factor each trinomial by guess and check.**

3x2 – 2x – 8 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 The factors of 3x2 – 2x – 8 are (x – 2)(3x + 4).

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**Example 1D: Factoring ax2 + bx + c**

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

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**Example 1E: Factoring ax2 + bx + c**

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

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** Example 1F Factor each trinomial. Check your answer. 6x2 + 17x + 5**

a = 6 and c = 5, Outer + Inner = 17. ( x + )( x + ) (3x + 1)(2x + 5) Use the Foil method. Check (3x + 1)(2x + 5) = 6x2 + 15x + 2x + 5 = 6x2 + 17x + 5

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** Example 1G Factor each trinomial. Check your answer. 9x2 – 15x + 4**

a = 9 and c = 4, Outer + Inner = –15. ( x + )( x + ) (3x – 4)(3x – 1) Use the Foil method. Check (3x – 4)(3x – 1) = 9x2 – 3x – 12x + 4 = 9x2 – 15x + 4

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** Example 1H Factor each trinomial. Check your answer. 3x2 + 13x + 12**

a = 3 and c = 12, Outer + Inner = 13. ( x + )( x + ) (x + 3)(3x + 4) Use the Foil method. Check (x + 3)(3x + 4) = 3x2 + 4x + 9x + 12 = 3x2 + 13x + 12

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**Example 1I: Factoring ax2 + bx + c**

Factor each trinomial. Check your answer. 3n2 + 11n – 4 a = 3 and c = – 4, Outer + Inner = 11 . ( n + )( n+ ) (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 1J: Factoring ax2 + bx + c**

Factor each trinomial. Check your answer. 2x2 + 9x – 18 a = 2 and c = –18, Outer + Inner = 9 . ( x + )( x+ ) (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 1K: Factoring ax2 + bx + c**

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

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Example 1L Factor each trinomial. Check your answer. 6x2 + 7x – 3 a = 6 and c = –3, Outer + Inner = 7. ( x + )( x+ ) (3x – 1)(2x + 3) Use the Foil method. Check (3x – 1)(2x + 3) = 6x2 + 9x – 2x – 3 = 6x + 7x – 3

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Example 1M Factor each trinomial. Check your answer. 4n2 – n – 3 a = 4 and c = –3, Outer + Inner = –1. ( x + )( x+ ) (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 2A: 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+ ) –1(x + 1)(2x + 3)

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Example 2B 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+ ) –1(2x + 3)(3x + 4)

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Example 2C 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+ ) –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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