Using Distributive Property to Factor Variable Expressions

How to use the distributive property to factor variable expressions, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

The distributive property works in two directions: you can use it to expand a product into a sum, or you can run it in reverse to factor a sum back into a product. Factoring with the distributive property means identifying the greatest common factor (GCF) of all terms and pulling it outside parentheses. This skill is tested directly on the GED Math exam and is the foundation of all algebra simplification.

What Is the Distributive Property?

The distributive property states: \(\color{blue}{a(b + c) = \text{ ab } + \text{ ac }}\).

Running it in reverse: \(\color{blue}{\text{ ab } + \text{ ac } = a(b + c)}\) — this is factoring. You identify the common factor \(\color{blue}{a}\) and pull it out, leaving the remaining terms inside parentheses.

How to Factor Using the Distributive Property

Step 1: Identify the GCF of all terms

Look at the coefficients and variables of every term. The GCF is the largest value that divides evenly into all of them.

• \(\color{blue}{12x + 8}\): GCF of 12 and 8 is \(\color{blue}{4}\).
• \(\color{blue}{15y – 10}\): GCF of 15 and 10 is \(\color{blue}{5}\).
• \(\color{blue}{6x^{2} + 9x}\): GCF is \(\color{blue}{3x}\) (common coefficient factor 3, common variable factor x).

Step 2: Divide each term by the GCF

These quotients become the terms inside the parentheses.

Step 3: Write the factored form

\(\color{blue}{\text{ GCF }(\frac{\text{ term }1}{\text{ GCF }} + \frac{\text{ term }2}{\text{ GCF }} + \ldots )}\)

Step 4: Verify by expanding

Distribute the GCF back through the parentheses and check you recover the original expression.

Step-by-Step Summary

1. List the terms of the expression.
2. Find the GCF of all coefficients and the lowest power of any shared variables.
3. Divide each term by the GCF.
4. Write the GCF outside parentheses and the quotients inside: GCF × (term 1 ÷ GCF + term 2 ÷ GCF + …).
5. Expand to verify correctness.

Watch: The Distributive Property Step-by-Step (Math with Mr. J)

Math with Mr. J explains both expanding and factoring using the distributive property:

Worked Examples

Example 1: Factor \(\color{blue}{12x + 8}\).

\(\color{blue}{\text{ GCF }(12, 8) = 4}\). Divide: \(\color{blue}{12x \div 4 = 3x}\), \(\color{blue}{8 \div 4 = 2}\).
Factored: \(\color{blue}{4(3x + 2)}\). Check: \(\color{blue}{4 \times 3x + 4 \times 2 = 12x + 8}\) ✓

Example 2: Factor \(\color{blue}{15y – 10}\).

\(\color{blue}{\text{ GCF }(15, 10) = 5}\). Divide: \(\color{blue}{15y \div 5 = 3y}\), \(\color{blue}{10 \div 5 = 2}\).
Factored: \(\color{blue}{5(3y – 2)}\). Check: \(\color{blue}{5 \times 3y – 5 \times 2 = 15y – 10}\) ✓

Example 3: Factor \(\color{blue}{6x^{2} + 9x}\).

\(\color{blue}{\text{ GCF } = 3x}\). Divide: \(\color{blue}{6x^{2} \div 3x = 2x}\), \(\color{blue}{9x \div 3x = 3}\).
Factored: \(\color{blue}{3x(2x + 3)}\). Check: \(\color{blue}{3x \times 2x + 3x \times 3 = 6x^{2} + 9x}\) ✓

Example 4: Factor \(\color{blue}{20n – 12n^{2}}\).

\(\color{blue}{\text{ GCF } = 4n}\). Divide: \(\color{blue}{20n \div 4n = 5}\), \(\color{blue}{12n^{2} \div 4n = 3n}\).
Factored: \(\color{blue}{4n(5 – 3n)}\). Check: \(\color{blue}{4n \times 5 – 4n \times 3n = 20n – 12n^{2}}\) ✓

Factoring Three Terms and Negative Common Factors

The same steps work when an expression has three or more terms. The GCF has to divide every term, not just two of them.

Example A: Factor 9 − 12x + 6y.

• The numbers are 9, 12 and 6. Their GCF is 3.
• Divide each term: 9 ÷ 3 = 3, 12x ÷ 3 = 4x, 6y ÷ 3 = 2y.
• Factored form: 3(3 − 4x + 2y).
• Check: 3 × 3 − 3 × 4x + 3 × 2y = 9 − 12x + 6y.

Example B: Factor 24a − 18b.

• The GCF of 24 and 18 is 6. There is no shared variable, since one term has a and the other has b.
• Factored form: 6(4a − 3b).

Example C: Factor −10x − 15 by pulling out a negative.

• The GCF of 10 and 15 is 5. Because the first term is negative, you can factor out −5 instead.
• −10x ÷ (−5) = 2x and −15 ÷ (−5) = 3.
• Factored form: −5(2x + 3).

Both 5(−2x − 3) and −5(2x + 3) are correct. Many teachers prefer the second form because the first term inside the parentheses is positive, which makes later steps easier.

Example D: Factor x(y + 2) + 3(y + 2). Here the common factor is the whole group (y + 2). Pull it out to get (y + 2)(x + 3). This idea is the basis of factoring by grouping in Algebra 1.

More Practice: Factoring Algebraic Expressions (Khan Academy)

Khan Academy provides additional factoring examples with a focus on identifying the GCF:

Common Mistakes When Factoring Out the GCF

Most errors in this skill come from a few habits. Watch for these before you check your answer.

• Using a common factor that is not the greatest. 18m − 24 = 2(9m − 12) is true, but 9 and 12 still share a 3. The fully factored answer is 6(3m − 4).
• Forgetting the 1. When a term equals the GCF, it leaves 1 behind. 5x + 5 = 5(x + 1), not 5(x).
• Dropping a variable. In 4y² + 6y, both terms contain y, so the GCF is 2y, not 2. The answer is 2y(2y + 3).
• Taking too high a power. Use the lowest power that every term has. In x³ + x², the GCF is x², giving x²(x + 1).
• Sign slips with a negative GCF. If you pull out −3 from −6x + 9, every sign inside flips: −3(2x − 3).
• Skipping the check. Distribute your answer back out. If you do not get the original expression exactly, something is off.

A quick test for a full factorization: look inside the parentheses. If the terms there still share a number or variable, you can factor again.

Exercises

Factor each expression by pulling out the GCF.

1. \(\color{blue}{6a + 10}\)
2. \(\color{blue}{8b – 20}\)
3. \(\color{blue}{9x + 12}\)
4. \(\color{blue}{4y^{2} + 6y}\)
5. \(\color{blue}{18m – 24}\)
6. \(\color{blue}{10p^{2} + 15p}\)

Answers

1. \(\color{blue}{2(3a + 5)}\)
2. \(\color{blue}{4(2b – 5)}\)
3. \(\color{blue}{3(3x + 4)}\)
4. \(\color{blue}{2y(2y + 3)}\)
5. \(\color{blue}{6(3m – 4)}\)
6. \(\color{blue}{5p(2p + 3)}\)

Algebra I for Beginners 2026 The Ultimate Step by Step Guide to Acing Algebra I

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Practice Quiz: 10 Questions

Frequently Asked Questions

How do you factor variable expressions using the distributive property?

Find the greatest common factor (GCF) of all the terms, divide each term by it, and write the GCF outside parentheses with the quotients inside. For 12x + 8, the GCF is 4, so 12x + 8 = 4(3x + 2). Always check by distributing back: 4 × 3x + 4 × 2 = 12x + 8.

How do you use the distributive property to factor polynomials?

Look for a GCF that includes both numbers and variables. In 6x² + 9x, both terms share a factor of 3 and a factor of x, so the GCF is 3x. Dividing gives 6x² ÷ 3x = 2x and 9x ÷ 3x = 3, so 6x² + 9x = 3x(2x + 3). Pulling out the GCF is often the first step before other factoring methods.

How do you use the distributive property to simplify expressions?

Multiply the number outside the parentheses by every term inside, then combine like terms. For 3(2x − 5) + 4x, distribute to get 6x − 15 + 4x, then combine to get 10x − 15. Watch the signs carefully: a negative in front, as in −2(x − 3), gives −2x + 6.

How do you use the distributive property to multiply?

Break one factor into a sum that is easy to work with, multiply each part, and add the results. For 7 × 48, think 7 × (50 − 2) = 350 − 14 = 336. With variables it works the same way: 5(y + 4) = 5y + 20. Factoring is simply this process run in reverse.

How do you use the distributive property to remove parentheses?

Multiply the term outside by each term inside, keeping track of the signs. For 4(x + 2), you get 4x + 8. A minus sign in front acts like multiplying by −1, so −(3y − 7) becomes −3y + 7. On the GED Math test, removing parentheses this way is a common step when solving equations.

How do you factor out the greatest common factor of three terms?

Find the largest number and the lowest power of any variable that divide all three terms. Divide each term by that GCF and write the results inside parentheses. For 9 − 12x + 6y, the GCF is 3, so the answer is 3(3 − 4x + 2y). Distribute to check that you get the original expression back.

Can you factor out a negative number?

Yes. Factoring out a negative flips the sign of every term inside the parentheses. For −10x − 15, you can write −5(2x + 3). Both −5(2x + 3) and 5(−2x − 3) are equal, but the first is usually preferred because the leading term inside is positive.

What if the terms have no common factor?

If the only number that divides every term is 1 and no variable appears in all terms, the expression cannot be factored with the distributive property. For example, 3x + 7 has no common factor other than 1, so it stays as it is. Other methods, such as grouping, may still apply to longer expressions.

Is factoring the opposite of distributing?

Yes. Distributing turns a product into a sum, as in 4(3x + 2) = 12x + 8. Factoring goes the other way, turning 12x + 8 back into 4(3x + 2). Because they undo each other, you can always check a factoring answer by distributing it.

Related Topics

• Using Area Models to Factor Variable Expressions
• Use Properties to Write Equivalent Expressions
• Identify Equivalent Expressions
• Use Algebra Tiles to Identify Equivalent Expressions
• Identifying Expressions and Equations