Back to Lesson

Teacher Guide: Factoring Out the GCF

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

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Factoring. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Identify the greatest common factor of polynomial terms
  • Factor out the GCF from polynomials with two or more terms
  • Factor polynomials with multiple variables
  • Verify factoring by distributing
Prerequisites
  • • Understanding of factors and multiples
  • • Exponent rules (especially division)
  • • Distributive property
Discussion Starters
  • 1. Why is it helpful to factor out the GCF before trying other factoring methods?
  • 2. How can you check if you have completely factored out the GCF?
  • 3. What happens if you accidentally factor out only part of the GCF?
  • 4. When might you choose to factor out a negative GCF?
Common Misconceptions

The GCF of variables is the highest power

Remediation: Use a concrete example: What goes into both x2x^2 and x3x^3? It must be x2x^2 or lower. Just like GCF of 6 and 12 is 6, not 12.

You can only factor out numbers, not variables

Remediation: Show that variables follow the same rules as numbers. If all terms have xx, then xx is a common factor.

Differentiation Ideas

For Struggling Students:

  • • Start with only numerical GCFs (no variables)
  • • Use factor trees visually for each coefficient
  • • Provide GCF as a hint, have students complete the factoring

For On-Level Students:

  • • Factor polynomials with numerical and single-variable GCFs
  • • Include three-term polynomials
  • • Practice identifying when GCF = 1

For Advanced Students:

  • • Factor with multiple variables
  • • Factor negative GCFs
  • • Connect to factoring completely (GCF then other methods)
Standards Alignment
  • A-SSE.A.2 (CCSS.MATH.CONTENT.HSA.SSE.A.2)

    Use the structure of an expression to identify ways to rewrite it

  • A-SSE.B.3 (CCSS.MATH.CONTENT.HSA.SSE.B.3)

    Choose and produce an equivalent form of an expression to reveal properties

Lesson Resources
  • visualFactor Tree Tool

    Interactive factor tree to find GCF of numbers

  • activityGCF Matching Game

    Match polynomials with their factored forms

  • worksheetGCF Factoring Practice

    Graduated difficulty problems

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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

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.

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

What if there is no common factor?

If the GCF is 1, you cannot factor out a GCF. Move on to other factoring methods like grouping or using special patterns.

Should I factor out a negative GCF?

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.

How do I know I found the right GCF?

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

More in This Topic

Introduction to Factoring

Intermediate25 min

Factoring by Grouping

Intermediate25 min

Factoring Trinomials (a=1)

Intermediate25 min

Factoring Trinomials (General)

Advanced25 min