Using Distributive Property to Factor Variable Expressions

How to using 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 }}).

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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: (color{blue}{text{ GCF }(&text{ bull }; + &text{ bull }; pm &text{ bull };)}).
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}}) ✓

More Practice: Factoring Algebraic Expressions (Khan Academy)

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

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)})

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Frequently Asked Questions

What is factoring with the distributive property?

It is the reverse of expanding. Instead of multiplying a factor into a parenthesis, you identify the common factor in each term and pull it outside. The result is a product of the GCF and the remaining terms in parentheses.

What if no common factor exists?

If the GCF is 1, the expression is already fully factored (it is said to be “prime” with respect to common factor factoring). You cannot pull out anything meaningful.

Can variables be part of the GCF?

Yes. If every term contains a variable, that variable (at its lowest power appearing in any term) is part of the GCF. For example, in (color{blue}{4x^{3} + 6x^{2}}), the GCF is (color{blue}{2x^{2}}), giving (color{blue}{2x^{2}(2x + 3)}).

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