Factoring by Grouping - Weebly

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8-8 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) 5 2b4 1 10b2 2 8b3 2 40b 5 2b4 2 8b3 1 10b2 2 40b 3

Multiply b2 1 5 and b 2 4. Multiply by 2b. 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 4x(x2 1 3)(x 1 2)

3. 72z4 1 48z3 1 126z2 1 84z

6z(4z2 1 7)(3z 1 2) 5. 12f 3 2 36f 2 1 60f 2 180

12(f 2 1 5)(f 2 3) 7. 56m3 2 28m2 2 42m 1 21

7(4m2 2 3)(2m 2 1) 9. 60x3 2 90x2 2 30x 1 45

15(2x2 2 1)(2x 2 3) 11. 6r3 1 9r2 2 60r

3r(2r 2 5)(r 1 4)

2. 24y4 1 6y3 1 36y2 1 9y 3y(2y2 1 3)(4y 1 1)

4. 2e4 2 8e3 1 18e2 2 72e

2e(e2 1 9)(e 2 4) 6. 16g4 2 56g3 1 64g2 2 224g

8g(g2 1 4)(2g 2 7) 8. 40n4 2 60n3 2 50n2 1 75n

5n(4n2 2 5)(2n 2 3) 10. 12p5 1 8p4 1 18p3 1 12p2

2p2(2p2 1 3)(3p 1 2) 12. 20s6 2 50s5 2 30s4

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

Date

8-8

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?

V = 12x3 + 8x2 -15x

12x3 1 8x2 2 15x 5 x(12x2 1 8x 2 15) 5 x(12x2 1 18x 2 10x 2 15) 5 xf6x(2x 1 3) 2 5(2x 1 3)g

5 x(6x 2 5)(2x 1 3)

Factor out x, the GCF for all three terms.

ac is ?180 and b is 8. Break 8x into two terms that have a sum of 8x and a product of 2180x2.

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.

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) 5 x(12x2 1 8x 2 15) 5 12x3 1 8x2 2 15x 3

Multiply 6x 2 5 and 2x 1 3. Combine like terms. 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 = 12x3 + 34x2 +14x

V = 10x3 + 65x2 +105x

2x, 3x 1 7, 2x 1 1 15.

5x, 2x 1 7, x 1 3 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.

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