Factoring Out the GCF

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

Intermediate25 minLesson

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.Find the GCF of all terms (numbers AND variables)
2.Divide each term by the GCF
3.Write as: $\text{GCF} \times (\text{remaining terms})$

6x^2 + 9x = 3x(2x + 3)

The GCF is

3x

because:

$3$ is the largest number that divides both $6$ and $9$
$x$ is the highest power of $x$ common to both terms

Try it now

What is the GCF of

6x

and

9

Worked Examples

Factor: $8x + 12$

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 $4$

Check for common variables

First term: $8x$ has $x$ \nSecond term: $12$ has no $x$ → No common variable factor

The GCF is just the number

GCF = $4$

Divide each term by the GCF

$8x \div 4 = 2x$ \n $12 \div 4 = 3$ → $(2x + 3)$

Write the factored form

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

Answer: $4(2x + 3)$

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 $6x^2 + 9x$ , the GCF is $3x$ , not just $3$ .

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 $x^3$ and $x^2$ , use $x^2$ (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: $3x(2x + 3) = 6x^2 + 9x$ 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: $x^2 + 3x + 5$ cannot be factored by GCF.

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:

Prime factors of 12:

223

Prime factors of 18:

233

GCF

2 × 3 = 6

LCM

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.

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is the GCF of $6x$ and $9$ ?

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 $12x^2 + 18x$ square meters, factoring gives $6x(2x + 3)$ , showing the dimensions.

1Try It Yourself

A garden has area $15x^2 + 10x$ 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 $P = 100x - 20x^2$ , factoring gives $P = 20x(5 - x)$ , showing profit is zero when $x = 0$ or $x = 5$ .

2Try It Yourself

Revenue is given by $R = 50x - 5x^2$ 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 $\text{GCF}(\text{quotients})$
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.

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., $6$ in $6x^2$ )
Common factor: A factor shared by two or more terms

Formula Card

GCF Factoring Pattern

ax + bx = x(a + b)

Factor out the common factor $x$ from both terms

General GCF Pattern

ac + bc = c(a + b)

Factor out any common factor $c$ from all terms