Name
8-8
Class
Date
Reteaching
Factoring by Grouping
You can factor some higher-degree polynomials by grouping terms and factoring
out the GCF to find the common binomial factor. Make sure to factor out a
common GCF from all terms first before grouping.
Problem
What is the factored form of 2b4 2 8b3 1 10b2 2 40b?
2b4 2 8b3 1 10b2 2 40b 5 2b(b3 2 4b2 1 5b 2 20)
2b is the GCF of all four terms. Factor out 2b
from each term.
5 2bfb2(b 2 4) 1 5(b 2 4)g Group terms into pairs and look for the GCF of
each pair. b2 is the GCF of the first pair, and 5
is the GCF of the second pair.
5 2b(b2 1 5)(b 2 4)
b 2 4 is the common binomial factor. Use the
Distributive Property to rewrite the expression.
Multiply to check your answer.
2b(b2 1 5)(b 2 4) 5 2b(b3 1 5b 2 4b2 2 20)
Multiply b2 1 5 and b 2 4.
5 2b4 1 10b2 2 8b3 2 40b
Multiply by 2b.
5 2b4 2 8b3 1 10b2 2 40b 3
Reorder the terms by degree.
The factored form of 2b4 2 8b3 1 10b2 2 40b is 2b(b2 1 5)(b 2 4).
Exercises
Factor completely. Show your steps.
1. 4x4 1 8x3 1 12x2 1 24x
2. 24y4 1 6y3 1 36y2 1 9y
4x(x2 1 3)(x 1 2)
3y(2y2 1 3)(4y 1 1)
3. 72z4 1 48z3 1 126z2 1 84z
4. 2e4 2 8e3 1 18e2 2 72e
6z(4z2 1 7)(3z 1 2)
2e(e2 1 9)(e 2 4)
5. 12f 3 2 36f 2 1 60f 2 180
6. 16g4 2 56g3 1 64g2 2 224g
12(f 2 1 5)(f 2 3)
8g(g2 1 4)(2g 2 7)
7. 56m3 2 28m2 2 42m 1 21
8. 40n4 2 60n3 2 50n2 1 75n
7(4m2 2 3)(2m 2 1)
5n(4n2 2 5)(2n 2 3)
9. 60x3 2 90x2 2 30x 1 45
10. 12p5 1 8p4 1 18p3 1 12p2
15(2x2 2 1)(2x 2 3)
2p2(2p2 1 3)(3p 1 2)
11. 6r3 1 9r2 2 60r
12. 20s6 2 50s5 2 30s4
3r(2r 2 5)(r 1 4)
10s4(2s 1 1)(s 2 3)
Prentice Hall Algebra 1 • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
79
Name
Class
8-8
Date
Reteaching (continued)
Factoring by Grouping
Polynomials can be used to express the volume of a rectangular prism. They can
sometimes be factored into 3 expressions to represent possible dimensions of the
prism. The three factors are the length, width, and height.
Problem
The plastic storage container to the right has a volume of
12x3 1 8x2 2 15x. What linear expressions could
represent possible dimensions of the storage container?
12x3 1 8x2 2 15x 5 x(12x2 1 8x 2 15)
V = 12x3 + 8x2 -15x
Factor out x, the GCF for all three terms.
5 x(12x2 1 18x 2 10x 2 15)
ac is –180 and b is 8. Break 8x into two terms
that have a sum of 8x and a product of 2180x2 .
5 xf6x(2x 1 3) 2 5(2x 1 3)g
Group the terms into pairs and factor out the GCF
from each pair. The GCF of the first pair is 6x. The
GCF of the second pair is 25.
5 x(6x 2 5)(2x 1 3)
2x 1 3 is the common binomial term. Use the
Distributive Property to reorganize the factors.
Multiply to check your answer.
x(6x 2 5)(2x 1 3) 5 x(12x2 1 18x 2 10x 2 15)
Multiply 6x 2 5 and 2x 1 3.
5 x(12x2 1 8x 2 15)
Combine like terms.
5 12x3 1 8x2 2 15x 3
Multiply by x.
Possible dimensions of the storage container are x, 6x 2 5, and 2x 1 3.
Exercises
Find linear expressions for the possible dimensions of each rectangular prism.
13.
14.
V = 10x3 + 65x2 +105x
V = 12x3 + 34x2 +14x
2x, 3x 1 7, 2x 1 1
5x, 2x 1 7, x 1 3
15.
16.
V = 60x3 - 68x2 -16x
V = 12x3 - 15x2 -18x
4x, 5x 1 1, 3x 2 4
3x, 4x 1 3, x 2 2
Prentice Hall Algebra 1 • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
80