﻿ Simplifying Polynomials Worksheet - Page 2 | Problems & Solutions

# Simplifying Polynomials Worksheet - Page 2

Simplifying Polynomials Worksheet
• Page 2
11.
Find GCF of $a$4, 24$a$2, 12$a$.
 a. $a$ b. 12 c. 12$a$ d. $a$2

#### Solution:

a4 = a × a × a × a
[Write the factors of a4.]

24a2 = 12 × 2 × a × a
[Write the factors of 24a2.]

12a = 12 × a
[Write the factors of 12a.]

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

The GCF of a4, 24a2 and 12a is: a

12.
Factor $x$3 - 3$x$2 + 2$x$ completely.
 a. $x$($x$ - 2)($x$ + 1) b. $x$($x$ - 2)($x$ - 1) c. $x$($x$ + 2)($x$ - 1) d. None of the above

#### Solution:

x3 - 3x2 + 2x
[Original expression.]

The GCF of x3, 3x2 and 2x is x.

= x(x2 - 3x + 2)
[Use GCF to factor.]

= x(x - 2)(x - 1)
[Factor the trinomial.]

13.
Factor 2$x$3 + 4$x$2 - 16$x$ completely.
 a. 2$x$($x$ - 4)($x$ - 2) b. 2$x$($x$ + 4)($x$ - 2) c. 2$x$($x$ + 4)($x$ + 2) d. None of the above

#### Solution:

2x3 + 4x2 - 16x
[Original expression.]

= 2x(x2 + 2x - 8)
[Use GCF to factor.]

= 2x(x + 4)(x - 2)
[Factor the trinomial.]

14.
Factor 3$x$3 + 9$x$2 + 54$x$ completely.
 a. -3$x$($x$ + 6)($x$ + 3) b. -3$x$($x$ - 6)($x$ + 3) c. -3$x$($x$ - 6)($x$ - 3) d. None of the above

#### Solution:

-3x3 + 9x2 + 54x
[Original expression.]

= -3x(x2 - 3x - 18)
[Use GCF to factor.]

= -3x(x - 6)(x + 3)
[Factor the trinomial.]

15.
Factor - 3$x$3 + 9$x$2 + 30$x$ completely.
 a. -3$x$($x$ - 5)($x$ - 2) b. -3$x$($x$ + 5)($x$ - 2) c. -3$x$($x$ - 5)($x$ + 2) d. None of the above

#### Solution:

-3x3 + 9x2 + 30x
[Original expression.]

= -3x(x2 - 3x - 10)
[Factor out the GCF.]

The factors are 2 and -5
[2 + (-5) = -3 and 2 x (-5) = -10]

= -3x(x - 5)(x + 2)
[Factorize.]

16.
Factor - 2$x$2 - 6$x$ + 36 completely.
 a. -2($x$ - 6)($x$ - 3) b. -2($x$ + 6)($x$ - 3) c. -2($x$ + 6)($x$ + 3) d. None of the above

#### Solution:

-2x2 - 6x + 36
[Original expression.]

= -2(x2 + 3x - 18)
[Use GCF to factor.]

= -2(x + 6)(x - 3)
[Factor the trinomial.]

17.
Factor 3$x$3 - 9$x$2 - 54$x$ completely.
 a. 3$x$($x$ + 6)($x$ + 3) b. 3$x$($x$ - 6)($x$ + 3) c. 3$x$($x$ - 6)($x$ - 3) d. None of the above

#### Solution:

3x3 - 9x2 - 54x
[Original expression.]

= 3x(x2 - 3x - 18)
[Use GCF to factor.]

= 3x (x - 6) (x + 3)
[Factor the trinomial.]

18.
Factor: - 6$x$3 + 3$x$2 + 54$x$ - 27
 a. (-6$x$ + 3)($x$ - 3)($x$ - 3) b. (-6$x$ - 3)($x$ + 3)($x$ - 3) c. (-6$x$ + 3)($x$ + 3)($x$ - 3) d. None of the above

#### Solution:

-6x3 + 3x2 + 54x - 27
[Original expression.]

= (-6x3 + 3x2) + (54x - 27)
[Group terms.]

= x2(-6x + 3) - 9(-6x + 3)
[Factor each group.]

= (-6x + 3) (x2 - 9)
[Use distributive property.]

= (-6x + 3)(x + 3)(x - 3)
[Use the formula a2 - b2 = (a + b)(a - b).]

19.
Factor: - 5$x$3 - 6$x$2 + 20$x$ + 24
 a. (-5$x$ - 6)($x$ - 2)($x$ - 2) b. (-5$x$ + 6)($x$ + 2)($x$ - 2) c. (-5$x$ - 6)($x$ + 2)($x$ - 2) d. None of the above

#### Solution:

-5x3 - 6x2 + 20x + 24
[Original expression.]

= (-5x3 - 6x2) + (20x + 24)
[Group terms.]

= -x2(5x + 6) + 4(5x + 6)
[Factor each group.]

= (-5x - 6) (x2 - 4)
[Use distributive property.]

= (-5x - 6) (x + 2)(x - 2)
[Use the formula a2 - b2 = (a + b)(a - b).]

20.
Factor: 4$x$3 - 7$x$2 + 4$x$ - 7
 a. None of the above b. (4$x$ - 7)($x$2 - 1) c. (4$x$ - 7)($x$2 + 1) d. (4$x$ + 7)($x$2 + 1)

#### Solution:

4x3 - 7x2 + 4x - 7
[Original expression.]

= (4x3 - 7x2) + (4x - 7)
[Group terms.]

= x2(4x - 7) + 1(4x - 7)
[Factor each group.]

= (4x - 7)(x2 + 1)
[Use distributive property.]