Factoring Out the GCF

Learn to identify and factor out the greatest common factor from polynomial expressions.

Intermediate25 minLesson
Printable worksheet

Definition

Factoring out the GCF means rewriting a polynomial as a product of its greatest common factor and another polynomial.
To factor out the GCF:
  1. 1.Find the GCF of all terms (numbers AND variables)
  2. 2.Divide each term by the GCF
  3. 3.Write as: GCF×(remaining terms)\text{GCF} \times (\text{remaining terms})
6x2+9x=3x(2x+3)6x^2 + 9x = 3x(2x + 3)
The GCF is 3x3x because:
  • 33 is the largest number that divides both 66 and 99
  • xx is the highest power of xx common to both terms

Try it now

What is the GCF of 6x6x and 99?

Worked Examples

Factor: 8x+128x + 12
1

Find the GCF of the coefficients

Factors of 8: 1, 2, 4, 8\nFactors of 12: 1, 2, 3, 4, 6, 12\nGCF = 4 → GCF of numbers is 44
2

Check for common variables

First term: 8x8x has xx\nSecond term: 1212 has no xx → No common variable factor
3

The GCF is just the number

GCF = 44
4

Divide each term by the GCF

8x÷4=2x8x \div 4 = 2x\n12÷4=312 \div 4 = 3 → (2x+3)(2x + 3)
5

Write the factored form

8x+12=4(2x+3)8x + 12 = 4(2x + 3)

Common Mistakes

Forgetting to include the variable in the GCF

Why it's wrong: Students often find the GCF of just the numbers and forget to check for common variable factors.

Correct: Always check BOTH numbers AND variables. For 6x2+9x6x^2 + 9x, the GCF is 3x3x, not just 33.

Using the highest power instead of lowest

Why it's wrong: For variables, students sometimes take the highest exponent instead of the lowest.

Correct: The GCF uses the LOWEST power. For x3x^3 and x2x^2, use x2x^2 (what goes into both).

Not dividing all terms

Why it's wrong: Students sometimes forget to divide one of the terms by the GCF.

Correct: Check your work by distributing: 3x(2x+3)=6x2+9x3x(2x + 3) = 6x^2 + 9x confirms it's correct.

Stopping when GCF is 1

Why it's wrong: If terms share no common factor, students may think they made an error.

Correct: Some polynomials have GCF = 1. Example: x2+3x+5x^2 + 3x + 5 cannot be factored by GCF.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Factor Polynomials Using the Greatest Common Factor (GCF) | Step-by-Step Examples

by Understand The Math

Watch on YouTube

Interactive Visual

Factor Tree

Number:
Composite
Prime

Enter a number to see its factor tree and prime factorization.

Venn Diagram - GCF & LCM

Number A:
Number B:
121822, 33Only in ACommonOnly in B

Prime factors of 12:

223

Prime factors of 18:

233

GCF

6

2 × 3 = 6

LCM

36

2 × 2 × 3 × 3 = 36

How it works:

  • • GCF = product of common prime factors (center of diagram)
  • • LCM = product of all prime factors (entire diagram)

Find the greatest common factor using prime factorization.

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the GCF of 6x6x and 99?

Why It Matters

Factoring out the GCF is the first step in almost every factoring problem. It:
  • Simplifies expressions before further factoring
  • Reveals hidden patterns that aren't obvious at first
  • Makes solving equations easier by reducing complexity
  • Is essential for simplifying fractions with polynomials
Without this skill, more advanced factoring becomes much harder!

Real World Applications

Simplifying Area Formulas

Architects and engineers use factoring to simplify expressions for areas and volumes.

Example:

If a rectangular room has area 12x2+18x12x^2 + 18x square meters, factoring gives 6x(2x+3)6x(2x + 3), showing the dimensions.

1Try It Yourself

A garden has area 15x2+10x15x^2 + 10x square feet.

Factor to find the dimensions.

Step 1: Write the mathematical expression

Factor out the GCF:

Analyzing Profit Functions

Business analysts factor expressions to understand profit relationships.

Example:

If profit is P=100x−20x2P = 100x - 20x^2, factoring gives P=20x(5−x)P = 20x(5 - x), showing profit is zero when x=0x = 0 or x=5x = 5.

2Try It Yourself

Revenue is given by R=50x−5x2R = 50x - 5x^2 dollars.

Factor to analyze the revenue function.

Step 1: Write the mathematical expression

Factor out the GCF:

Key Takeaways

  • 1The GCF includes BOTH the largest common number AND the lowest common variable powers
  • 2To factor: find GCF, divide each term by it, write as GCF(quotients)\text{GCF}(\text{quotients})
  • 3Always verify by distributing back to check your answer
  • 4Factoring out the GCF is usually the FIRST step before other factoring methods

Frequently Asked Questions

If the GCF is 1, you cannot factor out a GCF. Move on to other factoring methods like grouping or using special patterns.
If the GCF is 1, you cannot factor out a GCF. Move on to other factoring methods like grouping or using special patterns.
Sometimes yes! If the leading term is negative, factoring out −1-1 (or a negative GCF) can make the remaining expression easier to work with.
Distribute your answer back out. If you get the original expression, you factored correctly. Also, check that the terms inside the parentheses have no remaining common factor.

Glossary

GCF (Greatest Common Factor)
The largest factor that divides all terms in an expression
Factor
To write an expression as a product of simpler expressions
Coefficient
The numerical part of a term (e.g., 66 in 6x26x^2)
Common factor
A factor shared by two or more terms

Formula Card

GCF Factoring Pattern

ax+bx=x(a+b)ax + bx = x(a + b)

Factor out the common factor $x$ from both terms

General GCF Pattern

ac+bc=c(a+b)ac + bc = c(a + b)

Factor out any common factor $c$ from all terms

Today's challengeOne problem, three tries

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