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Simplifying Polynomials Worksheet

Simplifying Polynomials Worksheet
  • Page 1
 1.  
Find the GCF of 40x3 and 48x2.
a.
3x2
b.
8x2
c.
8x
d.
13x2


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.


Correct answer : (2)
 2.  
Factor the GCF out of 28x3 - 35x2.
a.
5x2(4x + 7)
b.
7x2(4x + 5)
c.
5x2(4x - 7)
d.
7x2(4x - 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.]


Correct answer : (4)
 3.  
Factor 4x3 + 20x2 + 16x completely.
a.
4x(x2 - 5x + 4)
b.
x(x2 + 5x + 4)
c.
x(x2 - 5x + 4)
d.
4x(x2 + 5x + 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.]


Correct answer : (4)
 4.  
Factor: 5x3 - 6x2 - 45x + 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.]


Correct answer : (2)
 5.  
Factor: 3x3 + x2 - 3x - 1
a.
(3x - 1)(x + 1)(x - 1)
b.
(- 3x + 1)(x + 1)(x - 1)
c.
(3x + 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.]


Correct answer : (3)
 6.  
Factor the GCF out of 5x3 + 25x2 + 25x.
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.]


Correct answer : (3)
 7.  
Factor the GCF out of 27a3 + 81a.
a.
a(27a2 + 3)
b.
-27a(a2 + 3)
c.
3a(a2 + 27)
d.
27a(a2 + 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.]


Correct answer : (4)
 8.  
Find the GCF of 5a2, 45a, 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.


Correct answer : (4)
 9.  
Find the GCF of 2a3, 4a2, 16a5.
a.
3a2
b.
2a2
c.
a2
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


Correct answer : (2)
 10.  
Factor: - 2a2 - 8a - 12
a.
-2(a2 + 4a - 6)
b.
-2(a2 + 4a + 6)
c.
-2(a2 - 4a - 6)
d.
-4(a2 - 4a - 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.]


Correct answer : (2)

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