Factoring Trinomials of the Type ax2 + bx + c

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Factoring Trinomials of the Type ax2 + bx + c ALGEBRA 1 LESSON 7-6 Factor 20x2 + 17x + 3. 20x2 + 17x + 3 F O I L 1 • 20 1 • 3 + 1 • 20 = 23 1 • 3 1 • 1 + 3 • 20 = 61 3 • 1 factors of a factors of c 2 • 10 2 • 3 + 1 • 10 = 16 1 • 3 2 • 1 + 3 • 10 = 32 3 • 1 4 • 5 4 • 3 + 1 • 5 = 17 1 • 3 20x2 + 17x + 3 = (4x + 1)(5x + 3) 7-6

Factoring Trinomials of the Type ax2 + bx + c ALGEBRA 1 LESSON 7-6 Factor 3n2 – 7n – 6. 3n2 –7n –6 (1)(3) (1)(–6) + (1)(3) = –3 (1)(–6) (1)(1) + (–6)(3) = –17 (–6)(1) (1)(–3) + (2)(3) = 3 (2)(–3) (1)(2) + (–3)(3) = –7 (–3)(2) 3n2 – 7n – 6 = (n – 3)(3n + 2) 7-6

Factoring Trinomials of the Type ax2 + bx + c ALGEBRA 1 LESSON 7-6 Factor 18x2 + 33x – 30 completely. 18x2 + 33x – 30 = 3(6x2 + 11x – 10) Factor out the GCF. Factor 6x2 + 11x – 10. 6x2 + 11x –10 (2)(3) (2)(–10) + (1)(3) = –17 (1)(–10) (2)(1) + (–10)(3) = –28 (–10)(1) (2)(–5) + (2)(3) = –4 (2)(–5) (2)(2) + (–5)(3) = –11 (–5)(2) (2)(–2) + (5)(3) = 11 (5)(–2) 6x2 + 11x – 10 = (2x + 5)(3x – 2) 18x2 + 33x – 30 = 3(2x + 5)(3x – 2) Include the GCF in your final answer. 7-6

Step 1: 24h2 + 10h – 6 = 2(12h2 + 5h – 3) Factor out the GCF, 2. Factoring by Grouping ALGEBRA 1 LESSON 7-6 Factor 24h2 + 10h – 6. Step 1: 24h2 + 10h – 6 = 2(12h2 + 5h – 3) Factor out the GCF, 2. Step 2: 12 • –3 = –36 Find the product ac. Step 3: Factors Sum –2(18) = –36 –2 + 18 = 16 –3(12) = –36 –3 + 12 = 9 –4(9) = –36 –4 + 9 = 5 Find two factors of ac that have a sum b. Use mental math to determine a good place to start. Step 4: 12h2 – 4h + 9h – 3 Rewrite the trinomial. Step 5: 4h(3h – 1) + 3(3h – 1) Factor by grouping. (4h + 3)(3h – 1) Factor again. 24h2 + 10h – 6 = 2(4h + 3)(3h – 1) Include the GCF in your final answer. 7-6

Step 1: 3x(12x2 + 17x + 6) Factor out the GCF, 3x. Factoring by Grouping ALGEBRA 1 LESSON 7-6 A rectangular prism has a volume of 36x3 + 51x2 + 18x. Factor to find the possible expressions for the length, width, and height of the prism. Factor 36x3 + 51x2 + 18x. Step 1: 3x(12x2 + 17x + 6) Factor out the GCF, 3x. Step 2: 12 • 6 = 72 Find the product ac. Step 3: Factors Sum 4 • 18 4 + 18 = 22 6 • 12 6 + 12 = 18 8 • 9 8 + 9 = 17 Find two factors of ac that have sum b. Use mental math to determine a good place to start. 7-6

Step 4: 3x(12x2 + 8x + 9x + 6) Rewrite the trinomial. Factoring by Grouping ALGEBRA 1 LESSON 7-6 (continued) Step 4: 3x(12x2 + 8x + 9x + 6) Rewrite the trinomial. Step 5: 3x[4x(3x + 2) + 3(3x + 2)] Factor by grouping. 3x(4x + 3)(3x + 2) Factor again. The possible dimensions of the prism are 3x, (4x + 3), and (3x + 2). 7-6

Factoring Trinomials of the Type ax2 + bx + c ALGEBRA 1 LESSON 7-6 Factor each expression. 1. 3x2 – 14x + 11 2. 6t2 + 13t – 63 3. 9y2 – 48y – 36 (x – 1)(3x – 11) (2t + 9)(3t – 7) 3(3y + 2)(y – 6) 7-6