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FACTORING: GREATEST COMMON MONOMIAL FACTOR

MATH LESSON

FACTORING: GREATEST COMMON FACTOR

The first step in factoring any polynomial is to factor out the GCF — the biggest factor shared by all terms!

Before factoring any polynomial, always look for the Greatest Common Factor first. It is the foundation of ALL factoring!
ax + ay = a(x + y) where a = GCF
ax + ay
=
GCF
×
(x + y)
What is the GCF?

The Greatest Common Factor is the largest factor that divides ALL terms evenly. It has two parts: a number part and a variable part.

Number part: GCF of all coefficients
Variable part: each common variable raised to the smallest exponent
Combine both parts to get the full GCF!
How to Find the GCF — Step by Step

Let's find the GCF of 12x³ and 18x²:

Step 1: GCF of numbers
12 = 2 × 2 × 3  |  18 = 2 × 3 × 3
Common: 2 × 3 = 6
Step 2: GCF of variables
x³  |  x²  → smallest exponent = 2
Variable GCF = x²
Step 3: Combine
GCF = 6x²
How to Factor Out the GCF

Once you have the GCF, divide each term by it and write the GCF outside the parentheses:

12x³ ÷ 6x² = 2x
18x² ÷ 6x² = 3
Write it out: 6x²(2x + 3)
Check: distribute 6x² back → 12x³ + 18x² ✓
Common Mistakes!

Mistake 1: Forgetting the variable part of the GCF.

6x² + 10x = 2(3x² + 5x) × WRONG
6x² + 10x = 2x(3x + 5) ✓ CORRECT

Mistake 2: Not getting the GREATEST common factor.

12x + 18 = 3(4x + 6) × NOT FULLY FACTORED
12x + 18 = 6(2x + 3) ✓ CORRECT (6 is the GCF, not 3)

Mistake 3: Wrong sign inside parentheses when factoring a negative GCF.

−4x + 8 = −4(x + 2) × WRONG
−4x + 8 = −4(x − 2) ✓ CORRECT (signs flip!)
Worked Examples
Example 1: Factor: 6x + 18
GCF of 6 and 18 = 6  |  No common variable
6x ÷ 6 = x  |  18 ÷ 6 = 3
→ 6(x + 3)
Example 2: Factor: 10x² + 5x
GCF of 10 and 5 = 5  |  GCF of x² and x = x
Full GCF = 5x
10x² ÷ 5x = 2x  |  5x ÷ 5x = 1
→ 5x(2x + 1)
Example 3: Factor: 12x³ − 6x²
GCF of 12 and 6 = 6  |  GCF of x³ and x² = x²
Full GCF = 6x²
12x³ ÷ 6x² = 2x  |  −6x² ÷ 6x² = −1
→ 6x²(2x − 1)
Example 4: Factor: 15x²y + 10xy
GCF of 15 and 10 = 5  |  GCF of x²y and xy = xy
Full GCF = 5xy
15x²y ÷ 5xy = 3x  |  10xy ÷ 5xy = 2
→ 5xy(3x + 2)
Example 5: Factor: 18x⁴ − 12x³ + 6x²
GCF of 18, 12, 6 = 6  |  GCF of x⁴, x³, x² = x²
Full GCF = 6x²
18x⁴ ÷ 6x² = 3x²  |  −12x³ ÷ 6x² = −2x  |  6x² ÷ 6x² = 1
→ 6x²(3x² − 2x + 1)

Remember: Always find the GREATEST common factor (not just any common factor), include variables, and check by distributing back!

LEARNING CHECK

Let's see what you learned!

Answer these 2 questions correctly to proceed to the Quiz. If wrong, questions will be reshuffled and you try again!
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MATH QUIZ

Quiz Time!

10 questions about Factoring using GCF. You got this!

Answer all questions first before submitting!
0/10

Made for students who want to master Factoring by GCF

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