1. Factoring by Grouping Find the GCF of the terms of each polynomial.
ALGEBRA 1 LESSON 9-8
(For help, go to Lessons 9-2 and 9-3.)
Find the GCF of the terms of each polynomial.
1. 6y2 + 12y – r3 + 15r2 + 21r
3. 30h3 – 25h2 – 40h 4. 16m3 – 12m2 – 36m
Find each product.
5. (v + 3)(v2 + 5) 6. (2q2 – 4)(q – 5)
7. (2t – 5)(3t + 4) 8. (4x – 1)(x2 + 2x + 3)
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2. Factoring by Grouping Solutions 1. 6y2 + 12y – 4 2. 9r3 + 15r2 + 21r
ALGEBRA 1 LESSON 9-8
Solutions
1. 6y2 + 12y – r3 + 15r2 + 21r
6y2 = 2 • 3 • y • y; 9r3 = 3 • 3 • r • r • r;
12y = 2 • 2 • 3 • y; 4 = 2 • 2; 15r2 = 3 • 5 • r • r; 21r = 3 • 7 • r;
GCF = 2 GCF = 3r
3. 30h3 – 25h2 – 40h 4. 16m3 – 12m2 – 36m
30h3 = 2 • 3 • 5 • h • h • h; 16m3 = 2 • 2 • 2 • 2 • m • m • m;
25h2 = 5 • 5 • h • h; 12m2 = 2 • 2 • 3 • m • m;
40h = 2 • 2 • 2 • 5 • h; 36m = 2 • 2 • 3 • 3 • m;
GCF = 5h GCF = 2 • 2 • m = 4m
5. (v + 3)(v2 + 5) = (v)(v2) + (v)(5) + (3)(v2) + (3)(5)
= v3 + 5v + 3v2 + 15
= v3 + 3v2 + 5v + 15
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3. Factoring by Grouping Solutions (continued) 6. (2q2 – 4)(q – 5)
ALGEBRA 1 LESSON 9-8
Solutions (continued)
6. (2q2 – 4)(q – 5)
= (2q2)(q) + (2q2)(–5) + (–4)(q) + (–4)(–5)
= 2q3 – 10q2 – 4q + 20
7. (2t – 5)(3t + 4)
= (2t)(3t) + (2t)(4) + (–5)(3t) + (–5)(4)
= 6t2 + 8t – 15t – 20
= 6t2 – 7t – 20
8. (4x – 1)(x2 + 2x + 3) = (4x)(x2) + (4x)(2x) + (4x)(3)
+ (–1)(x2) + (–1)(2x) + (–1)(3)
= 4x3 + 8x2 + 12x – x2 – 2x – 3
= 4x3 + (8 – 1)x2 + (12 – 2)x – 3
= 4x3 + 7x2 + 10x – 3
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4. Factoring by Grouping Factor 6x3 + 3x2 – 4x – 2.
ALGEBRA 1 LESSON 9-8
Factor 6x3 + 3x2 – 4x – 2.
6x3 + 3x2 – 4x – 2 = 3x2(2x + 1) – 2(2x + 1) Factor the GCF from each group of two terms.
= (2x + 1)(3x2 – 2) Factor out (2x + 1).
= 6x3 – 4x + 3x2 – 2 Use FOIL.
Check: 6x3 + 3x2 – 4x – (2x + 1)(3x2 – 2)
= 6x3 + 3x2 – 4x – 2 Write in standard form.
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5. Factoring by Grouping Factor 8t4 + 12t3 + 16t + 24.
ALGEBRA 1 LESSON 9-8
Factor 8t4 + 12t3 + 16t + 24.
8t4 + 12t3 + 16t + 24 = 4(2t4 + 3t3 + 4t + 6) Factor out the GCF, 4.
= 4[t3(2t + 3) + 2(2t + 3)] Factor by grouping.
= 4(2t + 3)(t3 + 2) Factor again.
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6. Factoring by Grouping Factor 24h2 + 10h – 6.
ALGEBRA 1 LESSON 9-8
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 – = 16
–3(12) = –36 – = 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.
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7. Factoring by Grouping
ALGEBRA 1 LESSON 9-8
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 • = 22
6 • = 18
8 • = 17
Find two factors of ac that have sum b. Use mental math to determine a good place to start.
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8. Factoring by Grouping (continued)
ALGEBRA 1 LESSON 9-8
(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).
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9. Factoring by Grouping Factor each expression. 1. 10p3 – 25p2 + 4p – 10
ALGEBRA 1 LESSON 9-8
Factor each expression.
1. 10p3 – 25p2 + 4p – 10
2. 36x4 – 48x3 + 9x2 – 12x
3. 16a3 – 24a2 + 12a – 18
(5p2 + 2)(2p – 5)
3x(4x2 + 1)(3x – 4)
2(4a2 + 3)(2a – 3)
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