# Factoring by GCF Worksheet

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

2.
Factor the GCF out of 10$x$3 - 15$x$2.
 a. 3$x$2(2$x$ - 5) b. 3$x$2(2$x$ + 5) c. 5$x$2(2$x$ + 3) d. 5$x$2(2$x$ - 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.]

3.
Factor the GCF out of 8$x$3 + 16$x$2 + 64$x$.
 a. 8$x$($x$2 + 2$x$ + 2) b. $x$(8$x$2 + 2$x$ + 8) c. 8$x$($x$2 + 2$x$ + 8) d. 2$x$($x$2 + 2$x$ + 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.]

4.
Factor: 5$x$3 + 40
 a. 5($x$3 + 6$x$2 + 12$x$ + 8) b. 5($x$ + 2)($x$2 + 2$x$ + 4) c. 5($x$3 - 6$x$2 + 12$x$ - 8) d. 5($x$ + 2)($x$2 - 2$x$ + 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).]

5.
Factor: $x$3 - 216
 a. ($x$ - 6)($x$2 + 6$x$ + 36) b. $x$3 + 18$x$2 + 108$x$ + 216 c. ($x$ - 6)($x$2 - 6$x$ + 36) d. $x$3 - 18$x$2 + 108$x$ - 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).]

6.
Find the GCF of 3$a$3, 9$a$2, 54$a$5.
 a. $a$2 b. 3$a$2 c. 1458$x$10 d. 4$a$2

#### 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

7.
Factor:
25$z$2 - 16$c$2
 a. (25$z$ - $c$)($z$ - 16$c$) b. (5$z$ + 4$c$)(5$z$ - 4$c$) c. (5$z$ - 4$c$)2 d. (5$z$ + 4$c$) 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).]

8.
Factor:
25$s$2 - 64$t$2
 a. (5$s$ + 8$t$) 2 b. (5$s$ + 8$t$) (5$s$ - 8$t$) c. (5$s$ - 8$t$) 2 d. (25$s$ - $t$)($s$ - 64$t$)

#### 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).]

9.
Factor:
4$x$2 + 12$x$ + 9
 a. (2$x$ + 3)2 b. (2$x$ - 3) (2$x$ + 3) c. (2$x$ - 3)2 d. (4$x$ + 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.]

10.
Factor:
9$x$2 - 24$\mathrm{xy}$ + 16$y$2
 a. (3$x$ + 4$y$) (3$x$ - 4$y$) b. (3$x$ - $y$)($x$ + 4$y$) c. (3$x$ + 4$y$)2 d. (3$x$ - 4$y$)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.]