 7.05 Factoring out the greatest common factor (GCF)

Lesson

Previously, we looked at how to use the distributive law to remove parentheses from algebraic expressions. Now we are going to look at this process in reverse so we can factor expressions (which means writing them with parentheses).

To factorize algebraic equations, we need to find the greatest common factors (GCF) between the terms. We need to consider both the numeric values and algebraic terms. We will start by looking at algebraic factors.

Factoring out a numeric factor

There are lots of ways to look for the greatest common factor between two numbers such as looking at the prime factorization or listing all factors. With lots of practice, you will be able to identify the GCF more quickly.

Worked examples

Question 1

Factor: $12x+20$12x+20

Think: The $x$x factor is not common between $12x$12x and $20$20, so we are looking for a numeric common factor. We need to consider factors of 12 and 20 to find the greatest common factor between $12x$12x and $20$20 is $4$4.

Factors of $12$12: $1,2,3,4,6,12$1,2,3,4,6,12

Factors of $20$20: $1,2,4,5,10,20$1,2,4,5,10,20

With the GCF of $4$4, we need to break $12x$12x and $20$20 into a product with $4$4.

$12x=4\left(3x\right)$12x=4(3x)

$20=4(5)$20=4(5)

Do:

$12x+20=4\left(3x+5\right)$12x+20=4(3x+5)

Reflect: We can always check to see if we have factored correctly by distributing it back out again.

 $4\left(3x+5\right)$4(3x+5) $=$= $4\times3x+4\times5$4×3x+4×5 $=$= $12x+20$12x+20 We got back the original expression, so we are correct!

Remember to consider how the operators will change if negative numbers are involved.

Question 2

Factor: $-100r-20$100r20

Think: There is no factor of r in the second term, so the GCF between $-100r$100r and $-20$20 must be numeric. $-20$20 is a factor of $-100$100, so the GCF is $-20$20.

$-100r=-20\times5r$100r=20×5r

$-20=-20\times1$20=20×1

Do: $-20\left(5r+1\right)$20(5r+1)

Reflect: When the common factor is the same as one of the terms, we can't forget to put the factor of $1$1 in the parentheses. We can always check to see if we have factored correctly by distributing it back out again.

 $-20\left(5r+1\right)$−20(5r+1) $=$= $-20\times5r-20\times1$−20×5r−20×1 $=$= $-100r-20$−100r−20 We got back the original expression, so we are correct!

Practice questions

Question 3

Factor the expression $6v+30$6v+30.

Question 4

Factor the expression $-8v^2+56$8v2+56.

Factoring out an algebraic factor

Identifying an algebraic factor

In addition to there being a numeric common factor, there may also be an algebraic common factor. In order to find the greatest common algebraic factor we can use the following steps.

Remember!
1. First, find the numerical GCF between the coefficients of the terms we are comparing.
2. Now, check if there are any variables that appear in both terms. If so, what is the least power of each variable that is present? The product of each variable raised to its least power will be the algebraic component of our greatest common factor.
3. Finally, multiply the numerical and algebraic components to form our greatest common algebraic factor.

Worked example

Question 5

What is the greatest common algebraic factor of $2x^2y$2x2y and $6xz$6xz?

Think: We need to look at the numeric coefficient first, then the algebraic variables, and then combine our findings into a solution.

Do: The numerical coefficients are $2$2 and $6$6, so the numerical GCF is $2$2.

The algebraic terms present in the first term are $x$x and $y$y, and those present in the second term are $x$x and $z$z. So $x$x is the only common variable between the two. The least power of $x$x present in both terms is $x^1=x$x1=x, so this will be the algebraic component of our greatest common algebraic factor.

Reflect: Our greatest common algebraic factor is $2x$2x.

The same process applies if we are asked to compare $3$3 or more terms, we just need to ensure any common numeric or algebraic variables are common to all terms.

Practice questions

Question 6

What is the greatest common algebraic factor of the terms $9ax$9ax and $8ba$8ba?

Question 7

Find the greatest common factor of $x^2y^4+x^5y^6$x2y4+x5y6.

Factoring with algebraic factors

We've already learned how to factor (or add parentheses to) expressions, but only when these expressions have common factors that are numbers. Now we are going to look at expressions that have both numbers and algebraic terms as factors.

Worked examples

Question 8

Factor: $a^2-3a$a23a

Think: There is no common numerical factor. There is a common algebraic factor. The greatest algebraic factor is $a$a. So the GCF is $a$a.

$a^2=a\left(a\right)$a2=a(a)

$-3a=-3\left(a\right)$3a=3(a)

Do:

$a^2-3a=a\left(a-3\right)$a23a=a(a3)

Question 9

Factor: $18w^4-27w^2$18w427w2

Think: The greatest common numerical factor is $9$9. The greatest common algebraic factor is $w^2$w2. So the GCF is $9w^2$9w2.

$18w^4=9w^2\left(2w^2\right)$18w4=9w2(2w2)

$-27w^2=9w^2\left(-3\right)$27w2=9w2(3)

Do:

$18w^4-27w^2=9w^2\left(2w^2-3\right)$18w427w2=9w2(2w23)

Practice questions

Question 10

Factor: $2u^2-8u$2u28u

question 11

Factor the following expression:

$z^2+4z^5$z2+4z5

Question 12

Factor the following expression by taking out the greatest common factor:

$9x^2y^2z-18xyz$9x2y2z18xyz

Question 13

Factor $9r^2t^2v^3+6r^3tv-12rt^3v^2$9r2t2v3+6r3tv12rt3v2.

Outcomes

A1.10.C

Determine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend

A1.10.D

Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property