Factoring Out a Common Factor

Q: What happens when a term is exactly the GCF?

It becomes $1$, and the $1$ must be written. In $5x + 5$, dividing the second term by the GCF gives $5 \div 5 = 1$, so the answer is $5(x + 1)$. Dropping the $1$ to write $5(x)$ loses a term — distribute to see it only gives back $5x$.

The distributive property multiplies a factor through parentheses: $3(2x + 5) = 6x + 15$ . Factoring runs that exact process in reverse. You start with $6x + 15$ , notice that both terms share a factor of $3$ , and pull it out front to get $3(2x + 5)$ . The value never changes — the two forms are equal for every $x$ — only the shape does.

The factor you pull out should be the greatest common factor, or GCF: the largest number that divides evenly into every term. Pull out anything smaller and the expression is only partly factored, with a common factor still hiding inside the parentheses.

Find the GCF

Look at the number in each term and ask: what is the largest number that divides evenly into all of them? For $6x + 15$ , the numbers are $6$ and $15$ . Both are divisible by $3$ , and no larger number works, so the GCF is $3$ .

When the coefficients are bigger, list a few factors of each and take the largest one they share. For $12x - 18$ : the factors of $12$ are $1, 2, 3, 4, 6, 12$ and the factors of $18$ are $1, 2, 3, 6, 9, 18$ . They share $2$ , $3$ , and $6$ — the GCF is $6$ .

Divide each term and write the GCF out front

Divide every term by the GCF and put what's left inside parentheses, with the GCF multiplied on the outside. For $6x + 15$ with a GCF of $3$ : $6x \div 3 = 2x$ and $15 \div 3 = 5$ , so $6x + 15 = 3(2x + 5)$ .

The sign between the terms carries straight into the parentheses. For $8x - 12$ with a GCF of $4$ : $8x \div 4 = 2x$ and $12 \div 4 = 3$ , and the minus sign stays put, giving $4(2x - 3)$ .

Check by distributing

Factoring and distributing are opposite moves, so distributing your answer must rebuild the original expression. Check $4(2x - 3)$ : multiply through to get $8x - 12$ ✓. If the check produces anything else, either the GCF or one of the divisions was off.

The check also exposes a partial factor. $2(6x - 9)$ multiplies back to $12x - 18$ correctly — but $6x$ and $9$ still share a factor of $3$ , so $2$ was not the greatest common factor. Fully factored, $12x - 18 = 6(2x - 3)$ . A finished answer has no common factor left inside the parentheses.

Worked examples

Example 1: factor a sum

Factor $6x + 15$ .

Find the GCF of

6

and

15

3

Divide each term by

3

6x \div 3 = 2x, \quad 15 \div 3 = 5

Write the GCF out front

3(2x + 5)

Check by distributing:

3(2x + 5) = 6x + 15

✓

Answer: $3(2x + 5)$

Example 2: keep the minus sign

Factor $8x - 12$ .

Find the GCF of

8

and

12

4

Divide each term by

4

8x \div 4 = 2x, \quad 12 \div 4 = 3

Write the GCF out front — the minus sign stays between the terms

4(2x - 3)

Answer: $4(2x - 3)$

Example 3: a term that is the GCF

Factor $5x + 5$ .

Find the GCF of

5

and

5

5

Divide each term by

5

— the second term becomes

1

, not nothing

5x \div 5 = x, \quad 5 \div 5 = 1

Write the GCF out front

5(x + 1)

Answer: $5(x + 1)$

Try one yourself

Common questions

What if I pull out a common factor that isn't the greatest?

The result is equal to the original but not fully factored. $12x - 18 = 2(6x - 9)$ is true, yet $6x$ and $9$ still share a $3$ . Look inside your parentheses when you finish — if the terms still share a factor, pull it out too. Fully factored: $6(2x - 3)$ .

What happens when a term is exactly the GCF?

It becomes $1$ , and the $1$ must be written. In $5x + 5$ , dividing the second term by the GCF gives $5 \div 5 = 1$ , so the answer is $5(x + 1)$ . Dropping the $1$ to write $5(x)$ loses a term — distribute to see it only gives back $5x$ .

How do I check a factored answer?

Distribute it. Multiplying the GCF back through the parentheses must reproduce the original expression exactly — same terms, same signs. $4(2x - 3) = 8x - 12$ confirms the factoring in one line.

Does the minus sign change anything?

No — it just rides along. The sign between the terms of the original expression appears between the terms inside the parentheses: $8x - 12 = 4(2x - 3)$ . Watch that it doesn't silently flip to a plus while you copy things down.

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Factoring Out a Common Factor

Find the GCF

Divide each term and write the GCF out front

Check by distributing

Worked examples

Example 1: factor a sum

Example 2: keep the minus sign

Example 3: a term that is the GCF

Try one yourself

Common questions

Keep learning

The Distributive Property

Combining Like Terms

Variables & Algebraic Expressions

Evaluating Expressions