How can I factor out a common factor from an expression?
Quick answer
Find a factor shared by every term, including any variable factors. Divide each term by that factor, place the quotients inside parentheses, and write the shared factor outside. Then multiply back to check: the expansion should match the original expression. For example, 6x + 15 = 3(2x + 5), and expanding the right side gives 6x + 15. [S1]
The question, the details, the next step.
What does it mean to factor out a common factor?
It means rewriting a sum as a product by taking a factor shared by all the terms and putting it outside parentheses. This reverses the distributive property: instead of multiplying a number into each term, you collect that number in front. For example, 4x + 12 becomes 4(x + 3). Multiplying 4 back into the parentheses restores the original expression. [S1]
The key phrase is “shared by all the terms.” A number or variable that appears in only some terms cannot be pulled out of the whole expression without changing it.
How do I find the factor, especially when the terms have variables?
Use this routine:
- Find the greatest common numerical factor of all the coefficients.
- Look for variables present in every term. For each shared variable, use the smallest exponent that appears.
- Multiply those shared parts together; that is the greatest common factor.
- Divide each original term by it, and write those quotients inside parentheses.
Example: 12x³ + 18x². The greatest common numerical factor of 12 and 18 is 6. Both terms have x, and the smaller power is x², so the common factor is 6x². Dividing gives 2x and 3, so the result is 6x²(2x + 3).
This is common-factor factoring: find the GCF shared by all terms, rewrite each term using that factor, then use distribution in reverse. [S1]
What if I’m unsure whether I divided each term correctly?
Multiply the outside factor by every term inside the parentheses. If you recover the original expression, the factoring checks out. For 6x + 15, factor out 3 to get 3(2x + 5). Expand: 3 · 2x + 3 · 5 = 6x + 15. The result matches the starting expression. Checking by multiplying back is a useful verification. [S1]
For the variable example, expand 6x²(2x + 3): the terms are 12x³ and 18x², which match the original. Be attentive to signs too: in 8y - 12, taking out 4 gives 4(2y - 3), not 4(2y + 3), because expansion must restore the subtraction.
Can you give me a reusable layout for doing this on homework?
Write the work in three lines:
Original: [expression] Common factor and quotients: [common factor]([first term ÷ factor] + or − [next term ÷ factor] + …) Check: expand the parentheses and compare term by term with the original.
Filled in for 15a + 25: Original: 15a + 25 Common factor and quotients: 5(3a + 5) Check: 5 · 3a + 5 · 5 = 15a + 25.
If no number or variable factor larger than 1 is shared by every term, there may be no nontrivial common factor to take out. This method establishes a common-factor factorization; it does not establish that an expression cannot be factored in some other way.
A practical checklist
List the terms and identify their numerical coefficients.
Find the greatest numerical factor shared by every coefficient.
Include only variables found in every term, using the smallest shared exponent.
Divide each term by the proposed common factor and preserve each sign.
Expand the result and confirm it reproduces the original expression.
A few more useful answers
Do I have to use the greatest common factor?
No. A smaller shared factor can produce a correct factorization, but using the greatest common factor usually pulls out as much as possible in one step.
What if only one term contains a variable?
That variable is not a common factor of the whole expression. A factor must divide every term.
How do I know when I have finished?
You have finished taking out the common factor when the terms inside the parentheses no longer share a nontrivial factor. This check is about common-factor factoring, not every possible kind of factoring.
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra 2e: Greatest Common Factor and Factor by Grouping openstax.org · Checked