# Factoring by GCF Worksheet - Page 3

Factoring by GCF Worksheet
• Page 3
21.
Factor:
x9c + 15 - 64y6d
 a. (x3c + 5 - 4y2d) [x6c + 10 + 4x3c - 5y2d - 16y4d] b. (x3c + 5 + 4y2d) [x6c + 10 + 4x3c - 5y2d - 16y4d] c. (x3c + 5 + 4y2d) [x6c + 10 + 4x3c + 5y2d + 16y4d] d. (x3c + 5 - 4y2d) [x6c + 10 + 4x3c + 5y2d + 16y4d]

#### Solution:

x9c + 15 - 64y6d

= x3(3c + 5) - 64y3(2d)

= (x3c + 5)3 - (4y2d)3
[Use power property amn = (am)n .]

= (x3c + 5 - 4y2d) [(x3c + 5)2 + 4x3c + 5y2d + (4y2d)2]
[a3 - b3 = (a - b) (a2 + ab + b2).]

= (x3c + 5 - 4y2d) [x6c + 10 + 4x3c + 5y2d + 16y4d]

22.
If the area of a square board is (81$z$2 + 144$z$ + 64) cm2, then find the length of its side.
 a. (9$z$ + 8) cm b. (10$z$ + 8) cm c. (9$z$ - 8) cm d. (9$z$ + 9) cm

#### Solution:

Area of the square board = 81z2 + 144z + 64

= (9z)2 + 2(9z) (8) + (8)2

= (9z + 8)2
[Use the formula: a² + 2ab + b² = (a + b)2.]

Area of the square board = (length)2

(length)2 = (9z + 8)2

So, the length is (9z + 8) cm.

23.
Express the volume 27$x$3 - 64 of box in factored form.
 a. (3$x$ - 4) (9$x$2 - 12$x$ + 16) b. (3$x$ + 4) (9$x$2 + 12$x$ + 16) c. (3$x$ - 4) (9$x$2 + 12$x$ - 16) d. (3$x$ - 4) (9$x$2 + 12$x$ + 16)

#### Solution:

Volume of the box = 27x3 - 64

= (3x)3 - (4)3

= (3x - 4)((3x)2 + (3x)(4) + (4)2)
[Use the formula: a³ - b³ = (a - b) (a2 + ab + b2).]

= (3x - 4)(9x2 + 12x + 16)
[Simplify.]

24.
Factor 3$a$$n$ + 9 + 4$a$$n$ + 4, assume all exponents are positive integers.
 a. $a$$n$ + 4[3$a$5 + 4] b. $a$$n$ + 4[3$a$5 + 3] c. $a$$n$ - 4[3$a$5 - 4] d. $a$$n$ - 4[3$a$5 + 4]

#### Solution:

3an + 9 + 4an + 4

= 3an + 4 + 5 + 4an + 4

= 3an + 4 · a5 + 4an + 4
[Use exponent property am + n = am. an.]

= an + 4[3a5 + 4]

25.
Factor:
$x$6$a$ $y$6b + 2$x$3$a$ $y$3bz4$a$ + $z$8$a$
 a. ($x$$a$ $y$3$b$ + $z$4$a$)2 b. ($x$3$a$$y$3$b$ + $z$4$a$)2 c. 2${x}^{9a}{y}^{9b}{z}^{12a}$ d. $x$3$a$ - $z$4$a$

#### Solution:

x6a y6b + 2x3a y3b z4a + z8a

= (x3a y3b)2 + 2 (x3a y3b) (z4a) + (z4a)2
[a = x3a y3b, b = z4a.]

= (x3ay3b + z4a)2
[(a + b)2.]

26.
A circle of radius $\sqrt{\frac{7}{22}}$ is to be cut from a square paper of side $x$. Express the area of the remaining portion in a factored form. [Take $\pi$ = $\frac{22}{7}$.]
 a. ($x$ + 1)($x$ - 1) b. $x$2 + 1 c. ($x$ - 10)2 d. ($x$ + 10)2

#### Solution:

Area of the remaining portion = Area of the square - Area of the circle

Area of the square = x2

Area of the circle = πr2 = 227(722)2 = 1
[Simplify.]

Area of the remaining portion = x2 - 1
[a = x, b = 1.]

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

27.
A small square piece of side $y$ is removed from a square cloth of side $x$ as shown. What is the area of the remaining portion?

 a. $x$2 + $y$2 b. ($x$ + $y$)2 c. ($x$ - $y$)2 d. ($x$ + $y$)($x$ - $y$)

#### Solution:

Area of big square cloth = x2

Area of removed cloth = y2

Area of the remaining cloth = x2 - y2

= (x + y)(x - y)
[a2 - b2 = (a + b)(a - b).]