# Factoring Polynomials Word Problems

Factoring Polynomials Word Problems
• Page 1
1.
Factor:
$a$($a$ - 8) - $b$($b$ - 8)
 a. ($a$ - $b$)($a$ - 8)($b$ + 8) b. ($a$ - $b$)($a$ + $b$ - 8) c. ($a$ - $b$)($a$ - 8)($b$ - 8) d. ($a$ - $b$)($a$ - $b$ - 16)

#### Solution:

a(a - 8) - b(b - 8)

= a2 - 8a - b2 + 8b

= a2 - b2 - 8a + 8b
[Group the terms.]

= (a - b)(a + b) - 8(a - b)

= (a - b)(a + b - 8)

2.
Factor:
$\frac{{x}^{3}}{7}$ - $\frac{4}{7}$$x$2 + $\frac{3x}{7}$
 a. $\frac{x}{7}$($x$ + 2)($x$ - 2) b. $\frac{x}{7}$($x$ - 2)($x$ - 2) c. $\frac{x}{7}$($x$ + 1)($x$ - 3) d. $\frac{x}{7}$($x$ - 1)($x$ - 3)

#### Solution:

x37 - 4 / 7x2 + 3x7

= x7 (x2 - 4x + 3)
[Use the GCF to factor.]

= x7(x2 - 3x - x + 3)

= x7 (x - 1)(x - 3)

3.
Factor 9$x$6$a$ - 81$y$6$b$ completely.
 a. 9 ($x$3$a$ - 3$y$3$b$) ($x$3$a$ + 3$y$3$b$) b. 9($x$3$a$ - 3$y$3$b$)2 c. 9($x$3$a$ + 3$y$3$b$)2 d. 9($x$6$a$ - 9$y$6$b$)

#### Solution:

9x6a - 81y6b

= 9(x6a - 9y6b)
[GCF is 9.]

= 9[(x3a)2 - (3y3b)2]
[Write as perfect squares.]

= 9(x3a - 3y3b) (x3a + 3y3b)
[a2 - b2 = (a + b)(a - b).]

4.
Factor:
2$d$$x$ + 8$d$ + 5$b$$x$ + 20$b$
 a. (2$\mathrm{dx + d}$)($\mathrm{bx}$ + 5$b$) b. (2$\mathrm{d + x}$) (5$b$ + 4) c. ($x$ + 5$b$) (2$d$ + 4) d. (2$d$ + 5$b$) ($x$ + 4)

#### Solution:

2dx + 8d + 5bx + 20b

= (2dx + 8d ) + (5bx + 20b)
[Group terms.]

= 2d(x + 4) + 5b (x + 4)
[Factor.]

= (2d + 5b) (x + 4)
[Factor again.]

5.
Factor:
169$a$2 + 130$\mathrm{ab}$ + 78$\mathrm{ca}$ + 30$\mathrm{cb}$ + 25$b$2
 a. (13$a$ + 5$c$) (13$a$ - 5$b$ - 6$c$) b. (13$a$ + 5$b$ + 6$c$)(6$c$ + 5$b$) c. (13$a$ + 5$b$) (13$a$ + 5$b$ + 6$c$) d. (13$a$ - 5$b$) (13$a$ + 5$b$ + 6$c$)

#### Solution:

= (169a2 + 130ab + 25b2) + (78ca + 30cb)
169a2 + 130ab + 78ca + 30cb + 25b2
[Rearrange.]

= (13a + 5b)2 + 6c(13a + 5b)
[(a + b)2 = a2 + 2ab + b2]

= (13a + 5b)(13a + 5b) + 6c(13a + 5b)

= (13a + 5b) (13a + 5b + 6c)
[Factor again.]

6.
Factor:
2$x$3 - 128
 a. 2($x$ - 4)($x$2 + 4$x$ + 16) b. ($x$ + 2)($x$2 - 4$x$) c. ($x$ - 4)($x$2 + 64) d. ($x$ - 4)(2$x$2 + 4$x$ + 16)

#### Solution:

= 2(x3 - 64)
2x3 - 128
[GCF is 2.]

= 2(x - 4)(x2 + 4x + 16)
[ a3 - b3 = (a - b) (a2 + ab + b2 ).]

7.
Factor:
25$e$2 - 30$\mathrm{eg}$ - 9$g$ + 15$e$ + 9$g$2
 a. (5$e$ - 3$g$) (5$e$ - 9$g$) b. (5$e$ - 3$g$) (5$e$ - 3$g$ + 3) c. (5$e$ + 3$g$) (5$e$ - 3$g$) d. (5$e$ - 3$g$) (5$e$ - 9$g$ + 3)

#### Solution:

25e2 - 30eg - 9g + 15e + 9g2

= (25e2 - 30eg + 9g2 ) + (- 9g + 15e)
[Rearrange.]

= (5e - 3g)2 + 3(5e - 3g)
[(a + b)2 = a2 + 2ab + b2.]

= (5e - 3g) (5e - 3g + 3)
[Factor.]

8.
Factor:
5$x$2 - 125$y$2
 a. (5$x$ - 5$y$)($x$ - 25$y$) b. ($x$ - 5$y$) (5$x$ + $y$) c. (5$x$ - $y$) ($x$ + $y$) d. 5($x$ - 5$y$) ($x$ + 5$y$)

#### Solution:

= 5(x2 - 25y2 )
5x2 - 125y2
[GCF is 5.]

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

9.
Factor:
$n$3 + $r$3 + $n$2 - $r$2
 a. ($n$3 + $r$3) b. ($n$3 + $r$3)($n$2 - $r$2) c. ($n$ + $r$) ($n$2 - $n$$r$ + $n$ - $r$ + $r$2) d. ($n$ + $r$) ($n$2 - $n$$r$ + $r$2)

#### Solution:

n3 + r3 + n2 - r2

= (n3 + r3) + (n2 - r2)
[Group terms.]

= (n + r) (n2 - nr + r2) + (n + r) (n - r)
[ a3 + b3= (a + b) (a2 - ab + b2) and (a2 - b2) = (a + b) (a - b)]

= (n + r) (n2 - nr + r2 + n - r)
[Factor again.]

= (n + r) (n2 - nr + n - r + r2)
[Arrange in descending order of exponents.]

10.
Factor:
$c$3 + $c$2 - 16$c$ - 448
 a. ($c$3 - 8) ($c$2 + 448) b. ($c$ - 8) ($c$2 + 9$c$ + 56) c. ($c$ - 8) ($c$2 + 8$c$ + 64) d. ($c$3 + $c$2)(16$c$ - 448)

#### Solution:

c3 + c2 - 16c - 448

= c3 + c2 - 16c - 512 + 64
[Replace - 448 as (- 512 + 64).]

= (c3 - 512) + (c2 - 16c + 64)
[Rearrange.]

= (c - 8) (c2 + 8c + 64) + (c - 8)2
(a3 - b3) = (a - b) (a2 + ab + b2) and (a - b)2 = a2 - 2ab + b2.]

= (c - 8) (c2 + 8c + 64 + c - 8)
[Factor again.]

= (c - 8) (c2 + 9c + 56)
[Simplify.]