# Simplifying Polynomials Worksheet - Page 3

Simplifying Polynomials Worksheet
• Page 3
21.
Factor: 3$x$2 + 12$x$ + 3$x$ + 12
 a. (3$x$ - 3)($x$ + 4) b. (3$x$ + 3)($x$ + 4) c. (3$x$2 +3)($x$ - 4) d. None of the above

#### Solution:

3x2 + 12x + 3x + 12
[Original expression.]

= (3x2 + 12x) + (3x + 12)
[Group terms.]

= 3x(x + 4) + 3(x + 4)
[Factor each group.]

= (3x + 3)(x + 4)
[Use distributive property.]

22.
Factor: $x$3 + 216
 a. ($x$ + 6)($x$2 + 6$x$ + 36) b. ($x$ - 6)($x$2 - 6$x$ + 36) c. ($x$ + 6)($x$2 - 6$x$ + 36) d. None of the above

#### Solution:

x3 + 216
[Original expression.]

= x3 + 63
[Write the terms as the sum of cubes.]

= (x + 6)(x2 - 6x + 36)
[Use (a3 + b3)= (a + b)(a2 + b2 - a x b).]

23.
Factor: 3$x$3 + 375
 a. 3(($x$ - 5)($x$2 + 5$x$ + 25)) b. 3(($x$ + 5)($x$2 - 5$x$ + 25)) c. 3(($x$ + 5)($x$2 + 5$x$ + 25) d. None of the above

#### Solution:

3x3 + 375
[Original expression.]

3(x3 + 125)
[Factor out the GCF.]

= 3(x3 + 53)
[Write the terms inside the grouping symbols as the sum of cubes.]

= 3[(x + 5)(x2 - 5x + 25)]
[Use (a3 + b3) =(a + b)(a2 - a x b + b2).]

24.
Factor: $x$3 - 64
 a. $x$ - 4)($x$2 - 4$x$ + 16) b. ($x$ - 4)($x$2 + 4$x$ + 16) c. ($x$ + 4)($x$2 + 4$x$ + 16) d. None of the above

#### Solution:

x3 - 64
[Original expression.]

= x3 - 43
[Write the terms as the difference between cubes.]

= (x - 4)(x2 + 4x + 16)
[Use (a3 - b3)=(a - b) (a2 + a x b + b2).]

25.
The length, width and height of a rectangular prism are ($x$ + 4), ($x$ - 1) and $x$, respectively. If the volume of the rectangular prism is 12 cubic units, then find the dimensions of the rectangular prism.
 a. 7 units, 6 units, 2 units b. 6 units, 1 units, 2 units c. 5 units, 1 units, 3 units d. None of the above

#### Solution:

The formula for the volume of a rectangular prism is l x b x h, where l is its length, b is its width and h is its height.

The volume of the rectangular prism in the question = 12cubic units and its length, width and height are (x + 4), (x - 1) and x, respectively.

12 = x(x - 1)(x + 4)
[Write the equation.]

12 = x3 + 3x2 - 4x
[Multiply.]

0 = x3 + 3x2 - 4x - 12
[Subtract 12 from each side.]

0 = (x3 + 3x2) - (4x + 12)
[Group terms.]

0 = x2(x + 3) - 4(x + 3)
[Use GCF to factor each group.]

0 = (x2 - 4)(x + 3)
[Use distributive property.]

0 = (x + 3)(x + 2)(x - 2)
[Factor the difference between two squares.]

x = -3 or -2 or 2
[Evaluate x.]

Since the dimension cannot be a negative value, x = 2.

(x + 4) = 2 + 4 = 6 and (x - 1) = 2 - 1 = 1

So, the dimensions of the rectangular prism are 6 units,1 unit and 2 units.