﻿ Factoring by GCF Worksheet - Page 2 | Problems & Solutions

# Factoring by GCF Worksheet - Page 2

Factoring by GCF Worksheet
• Page 2
11.
Factor:
8$x$3 - 1331
 a. (2$x$ + 11) (4$x$2 - 22$x$ + 121) b. (2$x$ - 11) (4$x$2 + 22$x$ + 121) c. (2$x$ + 11) (121$x$2 + 22$x$ + 4) d. (2$x$ - 11)(2$x$ + 11)

#### Solution:

8x3 - 1331

= (2x)3 - (11)3
[a = 2x, b = 11.]

= (2x - 11) [(2x)2 + (2x)(11) + (11)2]
[(a3 - b3) = (a - b) (a2 + ab + b2).]

= (2x - 11) (4x2 + 22x + 121)
[Simplify.]

12.
Factor:
1000$x$6 + 729$y$3
 a. (10$x$2 + 9$y$) (100$x$4 - 90$x$2$y$ + 81$y$2) b. (100$x$6 + 81$y$2) c. (10$x$2 + 9$y$) (100$x$4 + 90$x$2$y$ + 81$y$2) d. (10$x$2 + 9$y$) (100$x$4 + 81$y$2)

#### Solution:

1000x6 + 729y3

= (10x2)3 + (9y)3
[a = 10x2, b = 9y.]

= (10x2 + 9y) [(10x2)2 - (10x2) (9y) + (9y)2]
[(a3 + b3) = (a + b) (a2 - ab + b2).]

= (10x2 + 9y) (100x4 - 90x2y + 81y2)
[Simplify.]

13.
Factor:
${x}^{2}$ + $\frac{1}{5}$$x$ + $\frac{1}{100}$
 a. ($x$ - 10)2 b. ($x$ + 10)2 c. ($x$ + $\frac{1}{10}$)2 d. ($x$ - $\frac{1}{10}$)2

#### Solution:

x2 + 1 / 5x + 1100

= (x2 + 1 / 5x + (1 / 10)2)

= [(x2 + 2(1 / 10)x + (1 / 10)2]
[a = x, b = 1 / 10.]

= (x + 1 / 10)2
[a2 + 2ab + b2 = (a + b)2.]

14.
Factor:
$\frac{1}{5}$$x$2 + $\frac{4}{5}$$x$ + $\frac{4}{5}$
 a. 5($x$ - 2)2 b. 5($x$ + 2)2 c. $\frac{1}{5}{\left(x+2\right)}^{2}$ d. $\frac{1}{5}{\left(x-2\right)}^{2}$

#### Solution:

1 / 5x2 + 4 / 5x + 4 / 5

= 15(x2 + 4x + 4)

= 15(x2+2(x)(2)+22)
[a = x, b = 2.]

= 1 / 5(x + 2)2
[a2 + 2ab + b2 = (a + b)2.]

15.
Factor:
5$x$3 - $\frac{1}{25}$
 a. b. 25(5$x$ + 1)[ c. 25(5$x$ - 1)[ d.

#### Solution:

5x3 - 1 / 25

= 1 / 25(125x3 - 1)

= 1 / 25((5x)3 - 13)
[a = 5x, b = 1.]

= 125(5x - 1)[(5x)2+(5x)(1)+12)]
[(a3 - b3) = (a - b) (a2 + ab + b2).]

= 125(5x - 1)[25x2 + 5x + 1]
[Simplify.]

16.
Factor:
4$x$2 - $\frac{1}{64}$
 a. 64(16$x$ + 1)(16$x$ - 1) b. $\frac{1}{64}$(16$x$ + 1)2 c. $\frac{1}{64}$(16$x$ + 1)(16$x$ - 1) d. $\frac{1}{64}$(16$x$ - 1)2

#### Solution:

4x2 - 1 / 64

= 164(256x2 - 1)

= 164[(16x)2 -12]
[a = 16x, b = 1.]

= 1 / 64(16x + 1)(16x - 1)
[a2 - b2 = (a + b) (a - b).]

17.
Factor:
0.25$x$2 - 0.1$\mathrm{xy}$ + 0.01$y$2
 a. $\frac{1}{0.01}$(5$x$ - $y$)2 b. $\frac{1}{0.01}$(5$x$ + $y$)2 c. 0.01(5$x$ - $y$)2 d. 0.01(5$x$ + $y$)2

#### Solution:

0.25x2 - 0.1xy + 0.01y2

= 0.01[25x2 - 10xy + y2]

= 0.01[(5x)2 - 2(5x)(y) + (y)2]
[a = 5x, b = y.]

= 0.01(5x - y)2
[a2 - 2ab + b2 = (a - b) 2.]

18.
Factor:
(3$x$ + 6$y$)2 - 25$z$2
 a. (3$x$ + 6$y$ + 5$z$)2 b. (3$x$ + 6$y$ - 5$z$)2 c. (3$x$ + $y$ + 25$z$) (3$x$ + $y$ - 25$z$) d. (3$x$ + 6$y$ + 5$z$) (3$x$ + 6$y$ - 5$z$)

#### Solution:

(3x + 6y)2 - 25z2

= (3x + 6y)2 - (5z)2
[a = 3x + 6y, b = 5z.]

= (3x + 6y + 5z) (3x + 6y - 5z)
[(a + b) (a - b).]

19.
Factor:
$r$2 - (7$s$ + 3$t$)2
 a. ($r$ - 7$s$ - 3$t$)2 b. ($r$ + 7$s$ + 3$t$) ($r$ + 7$s$ - 3$t$) c. ($r$ + 7$s$ + 3$t$) ($r$ - 7$s$ - 3$t$) d. ($r$ + 7$s$ + $t$) ($r$ - 7$s$ - $t$)

#### Solution:

r2 - (7s + 3t)2

= [r + (7s + 3t)] [r - (7s + 3t)]
[a2 - b2 = (a + b)(a - b).]

= (r + 7s + 3t) (r - 7s - 3t)

20.
Factor:
$x$6$a$ - $y$6$a$

 a. IV b. II c. III d. I

#### Solution:

x6a - y6a

= (x3a)2 - (y3a)2
[Use power property, amn = (am)n.]

= (x3a + y3a)(x3a - y3a)
[(a2 - b2) = (a + b) (a - b).]

= [(xa)3 + (ya)3)] [(xa)3 - (ya)3)]
[Use power property, amn = (am)n.]

= [(xa+ya)(x2a-xaya+y2a)] [(xa-ya)(x2a+xaya+y2a)]
[a3 + b3 = (a + b) (a2 - ab + b2) and a3 - b3 = (a - b) (a2 + ab + b2).]

= (xa+ya)(xa-ya)(x2a-xaya+y2a) (x2a+xaya+y2a)