﻿ Simplifying Polynomials Worksheet | Problems & Solutions Simplifying Polynomials Worksheet

Simplifying Polynomials Worksheet
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1.
Find the GCF of 40$x$3 and 48$x$2. a. 3$x$2 b. 8$x$2 c. 8$x$ d. 13$x$2

Solution:

40x3 = 5 × 8 × x × x × x
[Write the factors.]

48x2 = 6 × 8 × x × x
[Write the factors.]

GCF = 8 × x × x = 8x2
[Multiply the common factors.]

The GCF of 40x3 and 48x2 is 8x2.

2.
Factor the GCF out of 28$x$3 - 35$x$2. a. 5$x$2(4$x$ + 7) b. 7$x$2(4$x$ + 5) c. 5$x$2(4$x$ - 7) d. 7$x$2(4$x$ - 5)

Solution:

28x3 = 4 × 7 × x × x × x
[Write the factors.]

35x2 = 5 × 7 × x × x
[Write the factors.]

GCF = 7 × x × x = 7x2
[Multiply the common factors.]

28x3 - 35x2
[Original expression.]

= 7x2(4x - 5)
[Use distributive property to factor out the greatest common factor from each term.]

3.
Factor 4$x$3 + 20$x$2 + 16$x$ completely. a. 4$x$($x$2 - 5$x$ + 4) b. $x$($x$2 + 5$x$ + 4) c. $x$(x2 - 5x + 4) d. 4$x$($x$2 + 5$x$ + 4)

Solution:

Find the greatest common factor of 4x3, 20x2 and 16x.

4x3 = 4 x x x x x x
[Write the factors.]

20x2 = 4 x 5 x x x x
[Write the factors.]

16x = 4 x 4 x x
[Write the factors.]

GCF = 4 x x = 4x
[Multiply the common factors.]

4x3 + 20x2 + 16x
[Original expression.]

= 4x(x2 + 5x + 4)
[Use the distributive property to factor out the greatest common factor from each term.]

4.
Factor: 5$x$3 - 6$x$2 - 45$x$ + 54 a. (-5x - 6)(x + 3)(x - 3) b. (5x - 6)(x + 3)(x - 3) c. (5x - 6)(x + 3)(- x - 3) d. (5x + 6)(x + 3)(x - 3)

Solution:

(5x3 - 6x2) + (-45x + 54)
[Group the terms.]

= x2(5x - 6) + (-9)(5x - 6)
[Factor out the GCF of each group.]

= (5x - 6)(x2 - 9)
[Use distributive property.]

= (5x - 6)(x + 3)(x - 3)
[Use difference between two squares pattern to factor.]

5.
Factor: 3$x$3 + $x$2 - 3$x$ - 1 a. (3$x$ - 1)($x$ + 1)($x$ - 1) b. (- 3$x$ + 1)($x$ + 1)($x$ - 1) c. (3$x$ + 1)($x$ + 1)($x$ - 1) d. None of the above

Solution:

= (3x3 + x2) + (-3x - 1)
[Group the terms.]

= x2(3x + 1) + (-1)(3x + 1)
[Factor out GCF of each group.]

= (3x + 1)(x2 - 1)
[Use distributive property.]

= (3x + 1)(x + 1)(x - 1)
[Use difference between two squares pattern to factor.]

6.
Factor the GCF out of 5$x$3 + 25$x$2 + 25$x$. a. 5x(x2 + 5x + 5) b. x(5x2 + 5x + 5) c. 5x(x2 + 5x + 5) d. 5x(x2 + 5x + 5)

Solution:

Find the greatest common factor of 5x3, 25x2 and 25x.

5x3 = 5 x x x x x x
[Write the factors.]

25x2 = 5 x 5 x x x x
[Write the factors.]

25x = 5 x 5 x x
[Write the factors.]

GCF = 5 x x = 5x
[Multiply the common factors.]

5x3 + 25x2 + 25x
[Original expression.]

= 5x(x2 + 5x + 5)
[Use the distributive property to factor out the greatest common factor from each term.]

7.
Factor the GCF out of 27$a$3 + 81$a$. a. $a$(27$a$2 + 3) b. -27$a$($a$2 + 3) c. 3$a$($a$2 + 27) d. 27$a$($a$2 + 3)

Solution:

27a3 = 3 x 3 x 3 x a x a x a
[Write the factors of 27a3.]

81a = 3 x 3 x 3 x 3 x a
[Write the factors of 81a.]

The GCF of two or more numbers is the product of their common factors.

The GCF of 27a3 and 81a is 3 x 3 x 3 x a = 27a

27a3 + 81a
[Original expression.]

= 27a(a2 + 3)
[Use the distributive property to factor out the greatest common factor from each term.]

8.
Find the GCF of 5$a$2, 45$a$, 65. a. 13 b. 10 c. 9 d. 5

Solution:

5a2 = 5 × a × a
[Write the factors of 5a2.]

45a = 5 × 9 × a
[Write the factors of 45a.]

65 = 5 × 13
[Write the factors of 65.]

The GCF of two or more numbers is the product of their common factors.

The GCF of 5a2, 45a and 65 is: 5.

9.
Find the GCF of 2$a$3, 4$a$2, 16$a$5. a. 3$a$2 b. 2$a$2 c. $a$2 d. None of the above

Solution:

2a3 = 2 × a × a × a
[Write the factors of 2a3.]

4a2 = 2 × 2 × a × a
[Write the factors of 4a2.]

16a5 = 2 × 8 × a × a × a × a × a
[Write the factors of 16a5.]

The GCF of two or more numbers is the product of their common factors.

The GCF of 2a3, 4a2 and 16a5 is: 2 × a × a = 2a2

10.
Factor: - 2$a$2 - 8$a$ - 12 a. -2($a$2 + 4$a$ - 6) b. -2($a$2 + 4$a$ + 6) c. -2($a$2 - 4$a$ - 6) d. -4($a$2 - 4$a$ - 6)

Solution:

-2a2 = -1 x 2 x a x a
[Write the factors of -2a2.]

-8a = -1 x 2 x 2 x 2 x a
[Write the factors of -8a.]

-12 = -1 x 2 x 2 x 3
[Write the factors of -12.]

The GCF of two or more numbers is the product of their common factors.

The GCF of -2a2, -8a and -12 is -2.

-2a2 - 8a - 12 = -2(a2 + 4a + 6)
[Use the distributive property to factor out the greatest common factor from each term.]