Factoring Out the GCF
Learn to identify and factor out the greatest common factor from polynomial expressions.
Definition
- is the largest number that divides both and
- is the highest power of common to both terms
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Worked Examples
Factor:
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
Check for common variables
First term: has \nSecond term: has no → No common variable factor
The GCF is just the number
GCF = →
Divide each term by the GCF
\n →
Write the factored form
→
Answer:
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 , the GCF is , not just .
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 and , use (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: 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: cannot be factored by GCF.
Interactive Visual
Factor Tree
Enter a number to see its factor tree and prime factorization.
Venn Diagram - GCF & LCM
Prime factors of 12:
Prime factors of 18:
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.
Interactive Sandbox
Expression Calculator
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Practice Problems
16 problemsWhat is the GCF of and ?
Why It Matters
- 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
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 square meters, factoring gives , showing the dimensions.
A garden has area 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 , factoring gives , showing profit is zero when or .
Revenue is given by 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
- 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
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., in )
- Common factor
- A factor shared by two or more terms
Formula Card
GCF Factoring Pattern
Factor out the common factor $x$ from both terms
General GCF Pattern
Factor out any common factor $c$ from all terms