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8-4 Factoring ax2 + bx + c Warm Up Find each product. 1. (x – 2)(2x + 7) 2x2 + 3x – 14 2. (3y + 4)(2y + 9) 6y2 + 35y + 36 3. (3n – 5)(n – 7) 3n2 – 26n + 35 Find each trinomial. 4. x2 +4x – 32 (x – 4)(x + 8) 5. z2 + 15z + 36 (z + 3)(z + 12) 6. h2 – 17h + 72 (h – 8)(h – 9) Holt Algebra 1 8-4 Factoring ax2 + bx + c Warm Up Holt Algebra 1 8-4 Factoring ax2 + bx + c Objective Factor quadratic trinomials of the form ax2 + bx + c. Holt Algebra 1 8-4 Factoring ax2 + bx + c Find a pair of numbers that multiply to the first number, and add to the second. 1. Multiplies to 24; Adds to 10 6&4 2. Multiplies to -8; Adds to -2 -4 & 2 3. Multiplies to 15; Adds to 8 5&3 4. Multiplies to 21; Adds to -10 -7 & -3 5. Multiplies to -27; Adds to 6 9 & -3 6. Multiplies to -49; Adds to 0 7 & -7 Holt Algebra 1 7. Multiplies to 36; Adds to 13 9&4 8. Multiplies to -19; Adds to -18 -19 & 1 9. Multiplies to 6; Adds to -5 -3 & -2 10. Multiplies to -14; Adds to -5 -7 & 2 11. Multiplies to 16; Adds to 10 8&2 12. Multiplies to 15; Adds to 16 15 & 1 8-4 Factoring ax2 + bx + c 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 = c Product = a ( X+ )( x+ ) = ax2 + bx + c Sum of outer and inner products = b Holt Algebra 1 8-4 Factoring ax2 + bx + c The guess and check method is usually not the most efficient method of factoring a trinomial. Look at the product of (x + 3) and (x + 4). x2 12 (x + 3)(x +4) = x2 + 7x + 12 3x 4x The coefficient of the middle term is the sum of 3 and 4. The third term is the product of 3 and 4. Holt Algebra 1 8-4 Factoring ax2 + bx + c Holt Algebra 1 8-4 Factoring ax2 + bx + c Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c When c is Positive Factor each trinomial. Check your answer. 2x2 + 17x + 21 ( x+ )( x+ a = 2 and c = 21, ) Outer + Inner = 17. Two numbers that multiply to 42; Add to 17: 14 & 3 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: 2x GCF of 2nd group: 3 Holt Algebra 1 8-4 Factoring ax2 + bx + c Remember! When b is negative and c is positive, the factors of c are both negative. Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 + 6x + 5 ( x+ )( x+ a = 1 and c = 5, ) Outer + Inner = 6. Two numbers that multiply to 5; Add to 6: 5 & 1 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: x GCF of 2nd group: 1 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 2x2 + 9x - 18 ( x+ )( x+ a = 2 and c = -18, ) Outer + Inner = 9. Two numbers that multiply to -36; Add to 9: 12 & -3 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: 2x GCF of 2nd group: -3 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 3x2 - 16x + 16 ( x+ )( x+ a = 3 and c = 16, ) Outer + Inner = -16. Two numbers that multiply to 48; Add to -16: -12 & -4 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: -4 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 4x2 - 15x - 4 ( x+ )( x+ a = 4 and c = -4, ) Outer + Inner = 13. Two numbers that multiply to -16; Add to -15: -16 & 1 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: 4x GCF of 2nd group: 1 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 8x + 15 ( x+ )( x+ a = 1 and c = 15, ) Outer + Inner = -8. Two numbers that multiply to 15; Add to -8: -5 & -3 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: x GCF of 2nd group: -3 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 6x2 + 17x + 5 ( x+ )( x+ a = 6 and c = 5, ) Outer + Inner = 17. Two numbers that multiply to 30; Add to 17: 15 & 2 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: 1 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 5x + 6 ( x+ )( x+ a = 1 and c = 6, ) Outer + Inner = -5. Two numbers that multiply to 6; Add to -5: -2 & -3 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: x GCF of 2nd group: -3 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 3x2 + 11x - 4 ( x+ )( x+ a = 3 and c = -4, ) Outer + Inner = 13. Two numbers that multiply to -12; Add to 11: 12 & -1 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: -1 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 9x2 - 15x + 4 ( x+ )( x+ a = 9 and c = 4, ) Outer + Inner = -15. Two numbers that multiply to 36; Add to -15: -12 & -3 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: -1 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 3x2 + 13x + 12 ( x+ )( x+ a = 3 and c = 12, ) Outer + Inner = 13. Two numbers that multiply to 36; Add to 13: 9 & 4 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: 4 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor x2 + 10x + 21. Show that the original polynomial and the factored form have the same value for n = 0, 1, 2, 3, and 4. x2 + 10x + 21 a = 1 and c = 21, ( x + )( x + ) Outer + Inner = 10. Two numbers that multiply to 21; Add to 10: 7 & 3 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: x GCF of 2nd group: 3 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 4A Continued Evaluate the original polynomial and the factored form for n = 0, 1, 2, 3, and 4. y2 + 10y + 21 y (y + 7)(y + 3) y 0 (0 + 7)(0 + 3) = 21 0 02 + 10(0) + 21 = 21 1 (1 + 7)(1 + 3) = 32 1 12 + 10(1) + 21 = 32 2 (2 + 7)(2 + 3) = 45 2 22 + 10(2) + 21 = 45 3 (3 + 7)(3 + 3) = 60 3 32 + 10(3) + 21 = 60 4 (4 + 7)(4 + 3) = 77 4 42 + 10(4) + 21 = 77 The original polynomial and the factored form have the same value for the given values of n. Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 9 ( x+ )( x+ a = 1 and c = -9, ) Outer + Inner = 0. Two numbers that multiply to -9; Add to 0: -3 & 3 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: x GCF of 2nd group: 3 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 36 ( x+ )( x+ a = 1 and c = -36, ) Outer + Inner = 0. Two numbers that multiply to -36; Add to 0: -6 & 6 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: x GCF of 2nd group: 6 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 81 ( x+ )( x+ a = 1 and c = -81, ) Outer + Inner = 0. Two numbers that multiply to -81; Add to 0: -9 & 9 Break up the b term into the two numbers that you found. Factor by grouping; GCF of 1st group: x GCF of 2nd group: 9 Holt Algebra 1 8-4 Factoring ax2 + bx + c Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 + 25 Two numbers that multiply to 25; Add to 0: …..? THE KEY to this problem is having c be a negative number, not a positive Holt Algebra 1 8-4 Factoring ax2 + bx + c When the leading coefficient is negative, factor out –1 from each term before using other factoring methods. Holt Algebra 1 8-4 Factoring ax2 + bx + c Caution When you factor out –1 in an early step, you must carry it through the rest of the steps. Holt Algebra 1 8-4 Factoring ax2 + bx + c Factor. Check your answer. Holt Algebra 1