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Warm Up

Preview. Warm Up. California Standards. Lesson Presentation. Warm Up Find each product. 1. ( x – 2)(2 x + 7) 2. (3 y + 4)(2 y + 9) 3. (3 n – 5)( n – 7) Find each trinomial. 4. x 2 + 4 x – 32 5. z 2 + 15 z + 36 6. h 2 – 17 h + 72. 2 x 2 + 3 x – 14.

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. 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)

  3. California Standards 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.

  4. 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 or 1.

  5. 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

  6. To factor a trinomial like ax2 + 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.

  7. ( x + )( x + )  (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 Additional Example 1: Factoring ax2 + bx + c Factor 6x2 + 11x + 4. Check your answer. 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. Try integer factors of 6 for the coefficients and integer factors of 4 for the constant terms.

  8. Additional Example 1 Continued Factor 6x2 + 11x + 4. Check your answer. The factors of 6x2 + 11x + 4 are (3x + 4) and (2x + 1). 6x2 + 11x + 4 = (3x + 4)(2x + 1) (3x + 4)(2x + 1) = Check Use the FOIL method. 6x2+ 3x + 8x + 4  = 6x2 + 11x + 4 The product of the original trinomial.

  9. ( x + )( x + )  (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 Check It Out! Example 1a Factor each trinomial. Check your answer. 6x2 + 11x + 3 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. Try integer factors of 6 for the coefficients and integer factors of 3 for the constant terms.

  10. (3x + 1)(2x + 3) = Check Check It Out! Example 1a Continued Factor each trinomial. Check your answer. The factors of 6x2 + 11x + 3 are (3x + 1)(2x + 3). 6x2 + 11x + 3 = (3x + 1)(2x +3) Use the FOIL method. 6x2 + 9x + 2x + 3  = 6x2 + 11x + 3 The product of the original trinomial.

  11. ( x + )( x + )  (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 Check It Out! Example 1b Factor each trinomial. Check your answer. 3x2– 2x– 8 The first term is 3x2, so at least one variable term has a coefficient other than 1. The coefficient of the x2 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.

  12. (x– 2)(3x + 4) = Check Check It Out! Example 1b Factor each trinomial. Check your answer. 3x2– 2x– 8 The factors of 3x2– 2x –8are (x– 2)(3x + 4). 3x2– 2x –8= (x– 2)(3x + 4) Use the FOIL method. 3x2+ 4x– 6x – 8  The product of the original trinomial. = 3x2– 2x– 8

  13. Product = c Product = a Sum of outer and inner products = b ( X + )( x + ) =ax2+bx+c So, to factor ax2 + 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.

  14. Product = c Product = a Sum of outer and inner products = b 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 + ) =ax2+bx+c

  15. ( x + )( x + ) Factors of 2 Factors of 21Outer+Inner  1 and 21 1(21) + 2(1) = 23 1 and 2  21 and 1 1(1) + 2(21) = 43 1 and 2  3 and 7 1(7) + 2(3) = 13 1 and 2  7 and 3 1(3) + 2(7) = 17 1 and 2  = 2x2 + 17x + 21 Additional 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. Use the FOILmethod. (x + 7)(2x + 3) Check (x + 7)(2x + 3)= 2x2 + 3x + 14x + 21

  16. Remember! When b is negative and c is positive, the factors of c are both negative.

  17. ( x + )( x + ) 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 Additional 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– 4)(3x– 4) Use the FOIL method. Check(x– 4)(3x– 4) = 3x2– 4x– 12x + 16

  18. ( x + )( x + ) 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 Check It Out! Example 2a Factor each trinomial. Check your answer. 6x2 + 17x + 5 a = 6 and c = 5; Outer + Inner = 17. Use the FOIL method. (3x + 1)(2x + 5) Check(3x + 1)(2x + 5)= 6x2 + 15x + 2x + 5

  19. ( x + )( x + ) 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 Check It Out! Example 2b Factor each trinomial. Check your answer. 9x2– 15x + 4 a = 9 and c = 4; Outer + Inner = –15. (3x– 4)(3x– 1) Use the FOIL method. Check (3x– 4)(3x– 1) = 9x2– 3x– 12x + 4

  20. ( x + )( x + ) 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 Check It Out! Example 2c Factor each trinomial. Check your answer. 3x2 + 13x + 12 a = 3 and c = 12; Outer + Inner = 13. (x + 3)(3x + 4) Use the FOIL method. Check (x + 3)(3x + 4) = 3x2 + 4x + 9x + 12  = 3x2 + 13x + 12

  21. 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.

  22. ( n + )( n+ ) 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 Additional 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 + 4)(3n– 1) Use the FOIL method. Check (n + 4)(3n– 1) = 3n2 –n + 12n –4

  23. ( x + )( x+ ) Factors of 2 Factors of –18Outer+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)(2x– 3)  = 2x2 + 9x –18 Additional 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 . Use the FOIL method. Check(x + 6)(2x– 3) = 2x2– 3x + 12x –18

  24. ( x + )( x+ ) Factors of 4 Factors of – 4Outer+Inner  1 and 4 1(4)– 1(4) = 0 –1 and4  1(2) – 2(4) = –6 1 and 4 –2 and2  –4 and1 1(1)– 4(4) = –15 1 and 4 (x – 4)(4x + 1)  = 4x2– 15x–4 Additional 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. Use the FOIL method. Check(x– 4)(4x + 1) = 4x2 + x – 16x–4

  25. ( x + )( x+ ) Factors of 6 Factors of –3Outer+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  1(–2) + 3(3) = 7 2 and 3 3 and–1 (2x + 3)(3x –1) Check It Out! Example 3a Factor each trinomial. Check your answer. 6x2 + 7x– 3 a = 6 and c = –3; Outer + Inner = 7. Use the FOIL method. Check(2x + 3)(3x –1) = 6x2– 2x + 9x – 3  = 6x + 7x –3

  26. ( n + )( n+ ) Factors of 4 Factors of –3Outer+Inner  1 and 4 1(–3) + 1(4) = 1 1 and –3  1(3) – 1(4) = –1 1 and 4 –1 and3 (n– 1)(4n + 3) Check It Out! Example 3b Factor each trinomial. Check your answer. 4n2–n– 3 a = 4 and c = –3; Outer + Inner = –1. Use the FOIL method. Check (n– 1)(4n + 3) = 4n2 + 3n – 4n – 3  = 4n2–n – 3

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

  28. Caution! When you factor out –1 in an early step, you must carry it through the rest of the steps and into the answer.

  29. –1( x + )( x+ ) Factors of 2 Factors of 3Outer+Inner  1 and 2 3 and1 1(1) + 3(2) = 7  1 and 2 1(3) + 1(2) = 5 1 and 3 (x + 1)(2x + 3) Additional Example 4: 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 + 1)(2x + 3)

  30. –1( x + )( x+ ) 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) 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(2x + 3)(3x + 4) Check–1(2x + 3)(3x + 4) = –6x2– 8x – 9x – 12  = –6x2– 17x – 12

  31. –1( x + )( x+ ) Factors of 3 Factors of 10Outer+Inner  1 and 3 2 and5 1(5) + 3(2) = 11  1 and 3 1(2) + 3(5) = 17 5 and 2 (x + 5)(3x + 2) 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 + 5)(3x + 2) Check–1(x + 5)(3x + 2) = –3x2– 2x – 15x – 10  = –3x2– 17x – 10

  32. 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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