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Factoring by GCF Worksheet

Factoring by GCF Worksheet
  • Page 1
 1.  
Write 5y3 + 10y2 + 15y as a product of two factors.
a.
15y(2y2)
b.
5y(y2 + 2y + 3)
c.
5y(y2 + 5)
d.
5y(y2 - 2y + 10)


Solution:

5y3 + 10y2 + 15y

5y3 = 5 × y × y × y
[Write as prime factors.]

10y2 = 2 × 5 × y × y
[Write as prime factors.]

15y = 5 × 3 × y
[Write as prime factors.]

GCF = 5y
[Find the common factor.]

5y3 = 5y × y2, 10y2 = 5y × 2y and 15y = 5y × 3

(5y3 + 10y2 + 15y) = 5y(y2 + 2y + 3)
[Distributive property.]


Correct answer : (2)
 2.  
Factor the GCF out of 10x3 - 15x2.
a.
3x2(2x - 5)
b.
3x2(2x + 5)
c.
5x2(2x + 3)
d.
5x2(2x - 3)


Solution:

10x3 = 2 × 5 × x × x × x
[Write the factors.]

15x2 = 3 × 5 × x × x
[Write the factors.]

GCF = 5 × x × x = 5x2
[Multiply the common factors.]

10x3 - 15x2

= 5x2(2x - 3)
[Distributive property.]


Correct answer : (4)
 3.  
Factor the GCF out of 8x3 + 16x2 + 64x.
a.
8x(x2 + 2x + 2)
b.
x(8x2 + 2x + 8)
c.
8x(x2 + 2x + 8)
d.
2x(x2 + 2x + 8)


Solution:

Find the greatest common factor of 8x3, 16x2 and 64x.

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

16x2 = 2 × 8 × x × x
[Write the factors.]

64x = 8 × 8 × x
[Write the factors.]

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

8x3 + 16x2 + 64x

= 8x(x2 + 2x + 8)
[Use the distributive property to factor out the greatest common factor from each term.]


Correct answer : (3)
 4.  
Factor: 5x3 + 40
a.
5(x3 + 6x2 + 12x + 8)
b.
5(x + 2)(x2 + 2x + 4)
c.
5(x3 - 6x2 + 12x - 8)
d.
5(x + 2)(x2 - 2x + 4)


Solution:

5x3 + 40

= 5(x3 + 8)
[Factor.]

= 5(x3 + 23)
[Write the terms inside the grouping symbols as the sum of cubes.]

= 5[(x + 2)(x2 - 2x + 4)]
[Using (a3 + b3) = (a + b)(a2 - ab + b2).]


Correct answer : (4)
 5.  
Factor: x3 - 216
a.
(x - 6)(x2 + 6x + 36)
b.
x3 + 18x2 + 108x + 216
c.
(x - 6)(x2 - 6x + 36)
d.
x3 - 18x2 + 108x - 216


Solution:

x3 - 216

= x3 - 63
[Write the terms as the difference between cubes.]

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


Correct answer : (1)
 6.  
Find the GCF of 3a3, 9a2, 54a5.
a.
a2
b.
3a2
c.
1458x10
d.
4a2


Solution:

3a3 = 3 × a × a × a
[Write the factors of 3a3.]

9a2 = 3 × 3 × a × a
[Write the factors of 9a2.]

54a5 = 3 × 18 × a × a × a × a × a
[Write the factors of 54a5.]

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

The GCF of 3a3, 9a2 and 54a5 is: 3 × a × a = 3a2


Correct answer : (2)
 7.  
Factor:
25z2 - 16c2
a.
(25z - c)(z - 16c)
b.
(5z + 4c)(5z - 4c)
c.
(5z - 4c)2
d.
(5z + 4c) 2


Solution:

25z2 - 16c2

= (5z)2 - (4c)2
[Write as a2 - b2, a = 5z, b = 4c.]

= (5z + 4c)(5z - 4c)
[a2 - b2 = (a + b) (a - b).]


Correct answer : (2)
 8.  
Factor:
25s2 - 64t2
a.
(5s + 8t) 2
b.
(5s + 8t) (5s - 8t)
c.
(5s - 8t) 2
d.
(25s - t)(s - 64t)


Solution:

25s2 - 64t2

= (5s) 2 - (8t) 2
[Write as a2 - b2, a = 5s, b = 8t.]

= (5s + 8t) (5s - 8t)
[a2 - b2 = (a + b) (a - b).]


Correct answer : (2)
 9.  
Factor:
4x2 + 12x + 9
a.
(2x + 3)2
b.
(2x - 3) (2x + 3)
c.
(2x - 3)2
d.
(4x + 3)(x + 3)


Solution:

4x2 + 12x + 9

= (2x)2 + 2(2x)(3) + (3)2
[a = 2x, b = 3.]

= (2x + 3)2
[a2 + 2ab + b2 = (a + b)2.]


Correct answer : (1)
 10.  
Factor:
9x2 - 24xy + 16y2
a.
(3x + 4y) (3x - 4y)
b.
(3x - y)(x + 4y)
c.
(3x + 4y)2
d.
(3x - 4y)2


Solution:

9x2 - 24xy + 16y2

= (3x)2 - 2(3x) (4y) + (4y)2
[a = 3x, b = 4y.]

= (3x - 4y)2
[a2 - 2ab + b2 = (a - b)2.]


Correct answer : (4)

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