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Simplifying Polynomials Worksheet - Page 3

Simplifying Polynomials Worksheet
  • Page 3
 21.  
Factor: 3x2 + 12x + 3x + 12
a.
(3x - 3)(x + 4)
b.
(3x + 3)(x + 4)
c.
(3x2 +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.]


Correct answer : (2)
 22.  
Factor: x3 + 216
a.
(x + 6)(x2 + 6x + 36)
b.
(x - 6)(x2 - 6x + 36)
c.
(x + 6)(x2 - 6x + 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).]


Correct answer : (3)
 23.  
Factor: 3x3 + 375
a.
3((x - 5)(x2 + 5x + 25))
b.
3((x + 5)(x2 - 5x + 25))
c.
3((x + 5)(x2 + 5x + 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).]


Correct answer : (2)
 24.  
Factor: x3 - 64
a.
x - 4)(x2 - 4x + 16)
b.
(x - 4)(x2 + 4x + 16)
c.
(x + 4)(x2 + 4x + 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).]


Correct answer : (2)
 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.


Correct answer : (2)

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