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7-4. Factoring ax 2 + bx + c. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Algebra 1. Holt Algebra 1. Warm Up Find each product. 1. ( x – 2)(2 x + 7) Find each trinomial. 2. x 2 + 4x – 32. ESSENTIAL QUESTION.

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7-4

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  1. 7-4 Factoring ax2+ bx + c Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

  2. Warm Up Find each product. 1. (x – 2)(2x + 7) Find each trinomial. 2. x2 +4x– 32

  3. ESSENTIAL QUESTION How do you factor quadratic trinomials of the form ax2 + bx + c ?

  4. 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

  5. (2x + 4)(3x + 1) = 6x2 + 14x + 4   (1x + 4)(6x + 1) = 6x2 + 25x + 4 (3x + 4)(2x + 1) = 6x2 + 11x + 4  (1x + 2)(6x + 2) = 6x2 + 14x + 4  (1x + 1)(6x + 4) = 6x2 + 10x + 4 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 coefficient of the x2 term is 6. The constant term in the trinomial is 4. Try factors of 6 for the coefficients and factors of 4 for the constant terms.

  6. (1x + 3)(6x + 1) = 6x2 + 19x + 3  (1x + 1)(6x + 3) = 6x2 + 9x + 3  (2x + 1)(3x + 3) = 6x2 + 9x + 3  (3x + 1)(2x + 3) = 6x2 + 11x + 3 Example 2 Factor each trinomial by guess and check. 6x2 + 11x + 3 ( _ + _ )( _ + _ ) Write two sets of parentheses. ( x + _)( _ x + _) The coefficient of the x2 term is 6. The constant term in the trinomial is 3. Try factors of 6 for the coefficients and factors of 3 for the constant terms.

  7. (1x–1)(3x + 8) = 3x2 + 5x– 8  (1x–2)(3x + 4) = 3x2– 2x– 8  (1x–4)(3x + 2) = 3x2– 10x– 8  (1x–8)(3x + 1) = 3x2– 23x– 8 Example 3 Factor each trinomial by guess and check. 3x2– 2x– 8 ( _ + _ )( _ + _ ) Write two sets of parentheses. ( x + _)( _ x + _) The coefficient of the x2 term is 3. The constant term in the trinomial is –8. Try factors of 3 for the coefficients and factors of 8 for the constant terms.

  8. Factors of 3 Factors of 16Outer+Inner  1 and 3 –1 and –16 1(–16) + 3(–1) = –19  1( – 8) + 3(–2) = –14 1 and 3 – 2 and – 8  – 4 and – 4 1( – 4) + 3(– 4)= –16 1 and 3  = 3x2– 16x + 16 Example 4: Factoring ax2 + bx + c When c is negative 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

  9. Factors of 6 Factors of 5Outer+Inner  1 and 5 1(5) + 6(1) = 11 1 and 6  1 and 5 2(5) + 3(1) = 13 2 and 3  1 and 5 3(5) + 2(1) = 17 3 and 2  = 6x2 + 17x + 5 Example 5 6x2 + 17x + 5 ( _ + _ )( _ + _ ) a = 6 and c = 5, Outer + Inner = 17. ( x + _)( _ x + _) Use the Foil method. (3x + 1)(2x + 5) Check(3x + 1)(2x + 5)= 6x2 + 15x + 2x + 5

  10. Factors of 9 Factors of 4Outer+Inner  3(–4) + 3(–1) = –15 3 and 3 –1 and – 4  3(–2) + 3(–2) = –12 3 and 3 – 2 and – 2  – 4 and – 1 3(–1) + 3(– 4)= –15 3 and 3  = 9x2– 15x + 4 Example 6 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

  11. Factors of 3 Factors of 12Outer+Inner  1 and 3 1 and 12 1(12) + 3(1) = 15  2 and 6 1(6) + 3(2) = 12 1 and 3  3 and 4 1(4) + 3(3) = 13 1 and 3  = 3x2 + 13x + 12 Example 7 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

  12. Factors of 3 Factors of –4Outer+Inner  1(4) + 3(–1) = 1 1 and 3 –1 and 4  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  = 3n2 + 11n –4 Example: Factoring ax2 + bx + c When c is Negative 3n2 + 11n– 4 ( _ + _ )( _ + _ ) a = 3 and c = – 4, Outer + Inner = 11 . ( x + _)( _ x + _) (n + 4)(3n– 1) Use the Foil method. Check (n + 4)(3n– 1) = 3n2 –n + 12n –4

  13. 1 and 4 1(4) + 4(–1) = 0 –1 and 4  1(2) + 4(–2) = –6 1 and 4 –2 and2  –4 and1 1(1) + 4(–4) = –15 1 and 4 (x – 4)(4x + 1)  = 4x2– 15x–4 Example: Factoring ax2 + bx + c When b & c are Negative 4x2– 15x– 4 a = 4 and c = –4, Outer + Inner = –15. ( _ + _ )( _ + _ ) ( x + _)( _ x + _) Factors of 4 Factors of – 4Outer+Inner Use the Foil method. Check(x– 4)(4x + 1) = 4x2 + x – 16x–4

  14. Factors of 4 Factors of –3Outer+Inner  1 and 4 1(–3)+ 4(1) = 1 1 and –3  1(3) – 4(1) = – 1 1 and 4 –1 and3 (4n + 3)(n– 1) Example 4n2–n– 3 ( _ + _ )( _ + _ ) a = 4 and c = –3, Outer + Inner = –1. ( x + _)( _ x + _) Use the Foil method. Check (4n + 3)(n– 1) = 4n2– 4n + 3n – 3 = 4n2–n – 3

  15. Factors of -2 Factors of -3Outer+Inner  1 and -2 3 and -1 -1(1) + 3(-2) = -7  1 and -2 1(1)+ -2(-3) = 7 -3 and 1 (x - 3)(-2x + 1) Example: Factoring ax2 + bx + c When a is Negative –2x2 + 7x - 3. (_ x + )(_ x+ ) Now: a = 2 and c = -3; Outer + Inner = 5 ANS: (x - 3)(-2x + 1)

  16. Factors of 6 Factors of 12Outer+Inner  2 and 3 4 and3 2(3) + 3(4) = 18  2 and 3 2(4)+ 3(3) = 17 3 and 4 (2x + 3)(3x + 4) Example –6x2– 17x– 12 ( _ + _ )( _ + _ ) Factor out –1. -1( x + _)( _ x + _) a = 6 and c = 12; Everything positive! –1(6x2 + 17x + 12) –1(2x + 3)(3x + 4)

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

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