4.5 Common Factors

Leanne Thompson

4.5 Common Factors

In earlier lessons in this unit, we multiplied factors together to get a product. Now, we will be reversing this process—we will start with a product, then break it down into its factors. Breaking down a product into factors is called factoring.

Table 4.5.1
Multiplying	Factoring
$8\times7=56$	$56=8 \bullet 7$
$2x(x+3)=2x^2+6x$	$2x^2+6x=2x(x+3)$

Greatest Common Factor

To factor expressions such as $2x^2+6x$ , we need to identify the greatest common factor of the terms in the expression. The greatest common factor (GCF) is the largest expression that is a factor of all the expressions.

Table 4.5.2
To find the greatest common factor (GCF) of two or more expressions:
Step 1: Factor each coefficient into its prime factorization. Write all the variables with exponents in expanded form. Step 2: List all the factors, matching common factors in a column. In each column, circle the common factors. Step 3: Bring down the common factors that all the expressions share. Step 4: Multiply the factors.

Example 1

Find the greatest common factor of $27{x}^{3}$ and $18{x}^{4}$ .

Table 4.5.3
Steps	Solution
Factor each coefficient into its prime factorization and write the variables with exponents in expanded form. Circle the common factors in each column.	$27x^3=3 \bullet 3 \bullet 3 \bullet \text{x} \bullet \text{x} \bullet \text{x}$ $18x^4=2 \bullet 3 \bullet 3 \bullet \text{x}\bullet \text{x} \bullet \text{x} \bullet \text{x}$
Bring down the common factors.	$\text{GCF}=3 \bullet 3 \bullet \text{x} \bullet \text{x} \bullet \text{x}$
Multiply the factors.	$\text{GCF}=9x^3$ The GCF of $27{x}^{3}$ and $18{x}^{4}$ is $9{x}^{3}$ .

Practice 1

Find the greatest common factor of the following expressions.

a) $12{x}^{2},18{x}^{3}$

b) $4{x}^{2}y,6x{y}^{3}$

c) $9{m}^{5}{n}^{2},12{m}^{3}n$

d) $21{x}^{3},9{x}^{2},15x$

Factor the Greatest Common Factor from a Polynomial

In arithmetic, we have seen that it can be useful to represent a number in factored form. For example, we can write 12 as ( $2 \bullet 6$ ) or ( $3 \bullet 4)$ . Similarly, it can be useful to represent a polynomial in factored form. One way to factor a polynomial is to find the greatest common factor of all the terms, then factor it out. To factor out a GCF, we can use the Distributive Property, but “in reverse” as seen in the steps below.

Table 4.5.4
Distributive Property	Distributive Property “in Reverse”
If a, b, and c are real numbers, then $a(b+c)=ab+ac$	If a, b, and c are real numbers, then $ab+ac=a(b+c)$

Table 4.5.5
To factor the greatest common factor from a polynomial:
Step 1: Find the GCF of all the terms of the polynomial. Step 2: Rewrite each term as a product using the GCF. Step 3: Use the “reverse” Distributive Property to factor the expression. Step 4: Check by multiplying the factors.

Example 2

Factor: $4x+12$

Table 4.5.6
Steps	Solution
Find the GCF of all the terms of the polynomial.	$4x=2 \bullet 2 \bullet \text{x}$ $12=2 \bullet 3 \bullet 3$ $\text{GCF}=2 \bullet 2$ $\text{GCF}=4$
Rewrite each term as a product using the GCF.	$4x+12$ $4 \bullet x+4 \bullet 3$
Use the “reverse” Distributive Property to factor the expression.	$4(x+3)$
Check by multiplying the factors.	$4(x+3)$ $4 \bullet \text{x}+4 \bullet 3$ $4x+12$

Practice 2

Factor the following expressions.

a) $6a+24$

b) $5a+5$

c) $12x-60$

d) $5{x}^{3}-25{x}^{2}$

Practice 3

Factor the following expressions by removing the greatest common factor.

a) $4{y}^{2}+24y+28$

b) $21{x}^{3}-9{x}^{2}+15x$

c) $-8y-24$

d) $-6{a}^{2}+36a$

Practice 4

The area of a rectangle can be represented by the expression ${l}^{2}-6l$ , where $l=$ length.
a) Write an expression to represent the width of the rectangle.
b) Given that $l$ = 12, what is the area and width of the rectangle?

In some polynomials, we can factor out a binomial instead of a monomial. This type of factoring will be part of a process that we will use to factor quadratic trinomials in the form ax²+ bx + c (where a ≠ 1) in a later lesson.

Example 3

Factor: $5q\left(q+7\right)-6\left(q+7\right)$

Table 4.5.7
Steps	Solution
Factor the GCF, (q + 7).	$(q+7)(5q-6)$
Check by multiplying the factors.	$5q\left(q+7\right)-6\left(q+7\right)$

Practice 5

Factor the polynomials.

a) $x(x+3)+5(x+3)$

b) $3a(a-5)-2(a-5)$

c) $4c(6-c)+16(6-c)$

d) $2(b+5)-b(5+b)$

Factor by Grouping

When we have a polynomial with four terms, we can sometimes factor by a process called grouping. This method involves breaking the four terms into two separate groups, then removing the greatest common factor from each pair of terms. We will also use factoring by grouping to factor quadratic trinomials in the form ax²+ bx + c (where a ≠ 1).

Table 4.5.8
To factor by grouping:
Step 1: Group terms with common factors. Step 2: Factor out the common factor in each group. Step 3: Factor the common factor from the expression. Step 4: Check by multiplying the factors.

Example 4

Factor: ${x}^{2}+3x+2x+6$

Table 4.5.9
Steps	Solution
Group terms with common factors.	( ${x}^{2}+3x$ ) $+(2x+6$ )
Factor out the common factor in each group.	$x(x+3)+2(x+3)$
Factor the common factor from the expression.	$(x+2)(x+3)$
Check by multiplying the factors.	$(x+2)(x+3)$ ${x}^{2}+3x+2x+6$

Practice 6

Factor by grouping.

a) ${a}^{2}+5a+8a+40$

b) ${x}^{2}+3x-2x-6$

c) ${y}^{2}+4y-7y-28$

d) $xy+8y+3x+24$

Homework

Find the greatest common factor of the following expressions.

a) $16{y}^{2},24{y}^{3}$

b) $6a{b}^{4},8{a}^{2}b$

c) $20{z}^{3}, 35{z}^{4}$

d) $12{a}^{5}, 48{a}^{6}$

e) $12{x}^{3}y^{2}, 18{x}^{2}y^{3}$

f) $24{a}^{4}b^{2}, 36{a}^{3}b^{3}$

g) $15{x}^{2}, 30{x}^{3}y^{2}$

h) $20{p}^{5}q^{3}, 30{p}^{4}q^{4}$

i) $25{m}^{4},35{m}^{3},20{m}^{2}$

j) $14{x}^{3},70{x}^{2},105x$
In each case, find the greatest common factor of the following expressions.

a) $-18{x}^{2}y, -24{x}^{3}y^{2}$

b) $-12{a}^{3}b, 24{a}^{4}b^{2}$

c) $-30{z}^{4}, 45{z}^{5}$

d) $-14{a}^{5}, -42{a}^{6}$

e) $36{x}^{2}y^{3}, -72{x}^{3}y^{2}$

f) $-18{x}^{4}y^{2}, -54{x}^{3}y^{3}$
Factor the following expressions.

a) $2b+14$

b) $14x+14$

c) $12p+12$

d) $18u-36$

e) $30y-60$

f) $2{x}^{3}+12{x}^{2}$

g) $6{y}^{3}-15{y}^{2}$

h) $15{z}^{2}-15{z}^{5}$
Factor the following expressions by removing the greatest common factor.

a) $5{x}^{2}-25x+15$

b) $3{y}^{2}-12y+27$

c) $20{x}^{3}-10{x}^{2}+14x$

d) $24{y}^{3}-12{y}^{2}-20y$

e) $9x{y}^{2}+6{x}^{2}{y}^{2}+21{y}^{3}$

f) $3{p}^{3}-6{p}^{2}q+9p{q}^{3}$
Factor the following expressions.

a) $-16z-64$

b) $-9y-27$

c) $-4{b}^{2}+16b$

d) $-7{a}^{2}+21a$
Factor the greatest common factor for each polynomial.

a) $8m-8$

b) $4n-4$

c) $9n-63$

d) $45b-18$

e) $3{x}^{2}+6x-9$

f) $4{y}^{2}+8y-4$

g) $8{p}^{2}+4p+2$

h) $10{q}^{2}+14q+20$

i) $8{y}^{3}+16{y}^{2}$

j) $12{x}^{3}-10x$

k) $5{x}^{3}-15{x}^{2}+20x$

l) $8{m}^{2}-40m+16$

m) $12x{y}^{2}+18{x}^{2}{y}^{2}-30{y}^{3}$

n) $21p{q}^{2}+35{p}^{2}{q}^{2}-28{q}^{3}$

o) $-2x-4$

p) $-3b+12$
The area of a rectangle is represented by the expression $-16{t}^{2}+80t+4$ , where $4$ is the width.
a) Write an expression to represent the length of the rectangle.
b) Given that t = 2, what is the area and length of the rectangle?
Factor the polynomials.

a) $4m\left(m+3\right)-7\left(m+3\right)$

b) $8n\left(n-4\right)+5\left(n-4\right)$

c) $5x\left(x+1\right)+3\left(x+1\right)$

d) $3b\left(b-2\right)-13\left(b-2\right)$

e) $2x\left(x-1\right)+9\left(x-1\right)$

f) $6m\left(m-5\right)-7\left(m-5\right)$
Factor by grouping.

a) ${x}^{2}+7x+3x+21$

b) $ab+7b+8a+56$

c) ${x}^{2}+2x-5x-10$

d) $xy+2y+3x+6$

e) $uv-9u+2v-18$

f) ${b}^{2}+5b-4b-20$

g) ${p}^{2}+4p-9p-36$

h) ${m}^{2}+6m-12m-72$

i) ${x}^{2}+xy+5x+5y$

j) $3{x}^{3}-7{x}^{2}+6x-14$
The greatest common factor of 36 and 60 is 12. Explain what this means.
What is the GCF of ${y}^{4},{y}^{5}$ , and ${y}^{10}$ ? Write a general rule that tells you how to find the GCF of ${y}^{a},{y}^{b}$ and ${y}^{c}$ .

Answers

1.

a) $8{y}^{2}$	b) $2ab$	c) $5{z}^{3}$	d) $12{a}^{5}$	e) $6{x}^{2}y^{2}$
f) $12{a}^{3}b^{2}$	g) $15{x}^{2}$	h) $10{p}^{4}q^{3}$	i) $5{m}^{2}$	j) $7x$

2.

a) $-6{x}^{2}y$	b) $-12{a}^{3}b$	c) $-15{z}^{4}$
d) $-14{a}^{5}$	e) $36{x}^{2}y^{2}$	f) $-18{x}^{3}y^{2}$

3.

a) $2\left(b+7\right)$	b) $14\left(x+1\right)$	c) $12\left(p+1\right)$	d) $18\left(u-2\right)$
e) $30\left(y-2\right)$	f) $2{x}^{2}\left(x+6\right)$	g) $3{y}^{2}\left(2y-5\right)$	h) $15{z}^{2}\left(1-z^{3}\right)$

4.

a) $5\left({x}^{2}-5x+3\right)$	b) $3\left({y}^{2}-4y+9\right)$	c) $2x\left(10{x}^{2}-5x+7\right)$
d) $4y\left(6{y}^{2}-3y-5\right)$	e) $3{y}^{2}\left(3x+2{x}^{2}+7y\right)$	f) $3p\left({p}^{2}-2pq+3{q}^{3}\right)$

5.

a) $-16\left(z+4\right)$

b) $-9\left(y+3\right)$

c) $-4b\left(b-4\right)$

d) $-7a\left(a-3\right)$

6.

a) $8(m - 1)$	b) $4(n - 1)$	c) $9(n - 7)$	d) $9(5b - 2)$
e) $3(x^{2} + 2x - 3)$	f) $4(y^{2} + 2y - 1)$	g) $2(4p^{2} + 2p + 1)$	h) $2(5q^{2} + 7q + 10)$
i) $8y^{2}(y + 2)$	j) $2x(6x^{2} - 5)$	k) $5x(x^{2} - 3x + 4)$	l) $8(m^{2} - 5m + 2)$
m) $6y^{2}(2x + 3x^{2} - 5y)$	n) $7q^{2}(3p + 5p^{2} - 4q)$	o) $-2(x + 2)$	p) $-3(b - 4)$

7.

a) $-4{t}^{2}+20t+1$
b) A = 100 units², L = 25 units

8.

a) $\left(m+3\right)\left(4m-7\right)$	b) $\left(n-4\right)\left(8n+5\right)$	c) $\left(x+1\right)\left(5x+3\right)$
d) $\left(b-2\right)\left(3b-13\right)$	e) $\left(x-1\right)\left(2x+9\right)$	f) $\left(m-5\right)\left(6m-7\right)$

9.

a) $\left(x+7\right)\left(x+3\right)$	b) $\left(a+7\right)\left(b+8\right)$
c) $\left(x-5\right)\left(x+2\right)$	d) $\left(y+3\right)\left(x+2\right)$
e) $\left(u+2\right)\left(v-9\right)$	f) $\left(b-4\right)\left(b+5\right)$
g) $\left(p-9\right)\left(p+4\right)$	h) $\left(m+6\right)\left(m-12\right)$
i) $\left(x+y\right)\left(x+5\right)$	j) $\left({x}^{2}+2\right)\left(3x-7\right)$