7.5Factoring with the GCF
Find the greatest common factor of every term, divide it out, and write the expression as a product. Always check by distributing — and always check whether more factoring is possible.
Why This Matters
Factoring out the GCF is the first step in every factoring problem. Skipping it makes every other factoring method harder and often leads to incomplete answers. Mastering the GCF now makes trinomial factoring, the AC method, and solving quadratic equations significantly easier.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Quick Answer
How Do You Factor Using the GCF?
Find the greatest factor shared by every term, divide each term by that factor, and write the expression as the GCF multiplied by the remaining polynomial. Always check your answer by distributing the GCF back into the parentheses.
Learning Goals
- Identify the GCF of the coefficients and the GCF of the variables.
- Factor numerical GCFs, variable GCFs, and combined GCFs.
- Factor negative GCFs when the leading term is negative.
- Factor expressions with two, three, or four or more terms.
- Recognize when GCF = 1 and no GCF factoring is possible.
- Identify when the GCF is only the first step and more factoring remains.
- Check every answer by distributing the GCF back.
Key Vocabulary
Greatest Common Factor (GCF)
The largest factor shared by every term in an expression. Includes both the coefficient GCF and the variable GCF.
Factoring
Rewriting an expression as a product of simpler expressions. The reverse of distributing.
Factor
A number or expression that divides evenly into another. In 4x(3x − 2), both 4x and (3x − 2) are factors.
Completely Factored
An expression is completely factored when no factor can be factored further over the integers.
Prime Polynomial
A polynomial whose only factors are 1 and itself — it cannot be factored further over the integers.
Leading Term
The term with the highest degree. When it is negative, consider factoring out a negative GCF.
Key Concepts — Finding and Factoring the GCF
How to Find the GCF — Step by Step
List the factors of each coefficient. The GCF is the largest factor shared by all of them.
12x³ − 8x² + 4x → coefficients: 12, 8, 4 → GCF = 4
Use the smallest exponent of each variable that appears in every term.
12x³ − 8x² + 4x → x³, x², x¹ → smallest exponent = 1 → GCF includes x¹
Multiply the coefficient GCF and the variable GCF together.
12x³ − 8x² + 4x → GCF = 4x
Write the expression as GCF × (remaining polynomial). Each term inside the parentheses is the original term divided by the GCF.
12x³ − 8x² + 4x = 4x(3x² − 2x + 1)
Multiply the GCF back into the parentheses. You should get the original expression.
4x(3x² − 2x + 1) = 12x³ − 8x² + 4x ✓
GCF Variable Rules — Quick Reference
Use the smallest exponent
x⁵, x³, x² → GCF = x²
The variable must appear in every term.
Variable absent from one term → not in GCF
x³ + x² + 5 → no x in GCF (5 has no x)
If any term lacks the variable, it cannot be part of the GCF.
Multiple variables — handle each separately
15a³b² + 20a²b³ → a: min exp = 2; b: min exp = 2 → GCF = 5a²b²
Find the GCF for each variable independently.
GCF = 1 is always valid
x² + 3x + 7 → GCF = 1 (no common factor > 1)
When GCF = 1, the expression cannot be factored by this method.
Factoring a Negative GCF
When the leading term is negative, it is often cleaner to factor out a negative GCF. This makes the leading term inside the parentheses positive.
Example: −3x² + 6x
GCF of coefficients: GCF(3, 6) = 3
GCF of variables: x¹ (smallest exponent)
Factor out −3x (negative because leading term is negative):
−3x² + 6x = −3x(x − 2)
Check: −3x · x = −3x²; −3x · (−2) = +6x ✓
Sign rule when factoring out a negative:
- Every sign inside the parentheses flips.
- A positive term becomes negative; a negative term becomes positive.
- Always verify by distributing back.
GCF Is Only the First Step
After factoring out the GCF, always check whether the remaining polynomial can be factored further. An expression is completely factored only when no factor can be factored again.
Example: 2x² − 18
Step 1 — GCF = 2: 2(x² − 9)
⚠ Not completely factored yet — x² − 9 is a difference of squares.
Step 2 — Factor x² − 9: 2(x − 3)(x + 3)
✓ Completely factored: 2(x − 3)(x + 3)
Worked Examples
Factor: 6x² + 9x (numerical and variable GCF)
Coefficients: GCF(6, 9) = 3
Variables: x², x¹ → smallest exponent = 1 → GCF includes x
GCF = 3x
Divide each term: 6x² ÷ 3x = 2x; 9x ÷ 3x = 3
6x² + 9x = 3x(2x + 3)
Check: 3x · 2x = 6x²; 3x · 3 = 9x ✓
Factor: 12x³ − 8x² + 4x (three-term polynomial)
Coefficients: GCF(12, 8, 4) = 4
Variables: x³, x², x¹ → smallest exponent = 1 → GCF includes x
GCF = 4x
Divide each term: 12x³ ÷ 4x = 3x²; 8x² ÷ 4x = 2x; 4x ÷ 4x = 1
12x³ − 8x² + 4x = 4x(3x² − 2x + 1)
Check: 4x · 3x² = 12x³; 4x · (−2x) = −8x²; 4x · 1 = 4x ✓
Factor: 15a³b² + 20a²b³ (two-variable GCF)
Coefficients: GCF(15, 20) = 5
Variable a: a³, a² → smallest exponent = 2 → GCF includes a²
Variable b: b², b³ → smallest exponent = 2 → GCF includes b²
GCF = 5a²b²
Divide each term: 15a³b² ÷ 5a²b² = 3a; 20a²b³ ÷ 5a²b² = 4b
15a³b² + 20a²b³ = 5a²b²(3a + 4b)
Check: 5a²b² · 3a = 15a³b²; 5a²b² · 4b = 20a²b³ ✓
Factor: 8x³ − 12x² + 4x (three-term polynomial, verify completely factored)
Coefficients: GCF(8, 12, 4) = 4
Variables: x³, x², x¹ → smallest exponent = 1 → GCF includes x
GCF = 4x
Divide each term: 8x³ ÷ 4x = 2x²; 12x² ÷ 4x = 3x; 4x ÷ 4x = 1
8x³ − 12x² + 4x = 4x(2x² − 3x + 1)
Check remaining factor 2x² − 3x + 1: does it factor? Find p, q: multiply to 2, add to −3 → no integer pair works.
Check: 4x · 2x² = 8x³; 4x · (−3x) = −12x²; 4x · 1 = 4x ✓
The remaining trinomial 2x² − 3x + 1 does factor: (2x − 1)(x − 1). Completely factored: 4x(2x − 1)(x − 1)
Factor: −3x² + 6x (negative GCF)
Leading term is negative → factor out a negative GCF.
Coefficients: GCF(3, 6) = 3 → use −3
Variables: x², x¹ → smallest exponent = 1 → GCF includes x
GCF = −3x
Divide each term: −3x² ÷ (−3x) = x; 6x ÷ (−3x) = −2
−3x² + 6x = −3x(x − 2)
Check: −3x · x = −3x²; −3x · (−2) = +6x ✓
Factor completely: 2x² − 18 (GCF is only the first step)
Coefficients: GCF(2, 18) = 2
Variables: x², constant → constant has no x → GCF has no variable part
GCF = 2
2x² − 18 = 2(x² − 9)
⚠ Not completely factored — x² − 9 is a difference of squares.
Factor x² − 9: a = x, b = 3 → (x − 3)(x + 3)
2x² − 18 = 2(x − 3)(x + 3)
Check: 2(x − 3)(x + 3) = 2(x² − 9) = 2x² − 18 ✓
Common Mistakes
Choosing a common factor that is not the greatest — e.g., factoring out 2 instead of 4 from 12x + 8.
Always find the largest factor shared by all terms. GCF(12, 8) = 4, so factor out 4.
Forgetting the variable part of the GCF — e.g., writing GCF = 4 instead of 4x for 12x² + 8x.
The GCF includes both the coefficient GCF and the variable GCF. Check every term for shared variables.
Using the largest exponent instead of the smallest — e.g., writing GCF = x³ for x³ + x².
Use the smallest exponent of each variable that appears in every term. GCF(x³, x²) = x².
Changing signs incorrectly when factoring — e.g., writing 4x(3x + 2) for 12x² − 8x.
Divide each term carefully: 12x² ÷ 4x = 3x; −8x ÷ 4x = −2. Result: 4x(3x − 2).
Forgetting to factor out a negative GCF when the leading term is negative.
When the leading term is negative, factor out a negative GCF to make the expression inside the parentheses cleaner.
Leaving a common factor inside the parentheses — e.g., writing 2(4x + 6) instead of 2(2x + 3) for 4x + 6.
After factoring, check that the terms inside the parentheses share no common factor. GCF(4, 6) = 2, so factor out 2 fully: 2(2x + 3).
Failing to distribute to check — assuming the factored form is correct without verifying.
Always multiply the GCF back into the parentheses and confirm you get the original expression.
Stopping after the GCF when more factoring is possible — e.g., leaving 2(x² − 9) as the final answer.
After the GCF, check whether the remaining polynomial can be factored further. 2(x² − 9) = 2(x − 3)(x + 3).
Factoring a term that is not shared by every term — e.g., factoring x from x² + x + 5 because two terms have x.
The GCF must divide every term. Since 5 has no x, the variable x cannot be part of the GCF.
Writing only the inside polynomial and forgetting the GCF — e.g., writing (3x − 2) instead of 4x(3x − 2).
The GCF is a factor and must appear in the final answer. The factored form is always GCF × (remaining polynomial).
Guided Practice
Guided Practice Video: Factoring
Review GCF factoring, trinomial factoring, difference of squares, and perfect square trinomials before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Factor: 5x² + 15x
Hint: Find GCF of coefficients 5 and 15. Both terms have x — use the smallest exponent.
Factor: 18y³ − 12y²
Hint: GCF of 18 and 12? Both terms have y — what is the smallest exponent of y?
Factor: 6a²b + 9ab²
Hint: Find GCF of coefficients. Then find GCF of a terms and GCF of b terms separately.
Factor: −4x³ + 8x²
Hint: The leading term is negative. Factor out a negative GCF. What is GCF(4, 8)? What is the smallest exponent of x?
Factor: 10x⁴ − 15x³ + 5x²
Hint: GCF of 10, 15, 5? All three terms have x — use the smallest exponent.
Factor completely: 3x² − 75
Hint: Factor out the GCF first. Then check whether the remaining expression is a difference of squares.
Math Tips
Always check for a GCF before trying any other factoring method.
The GCF of the coefficients is found by listing factors or using prime factorization.
For variables, the GCF uses the smallest exponent — not the largest.
If a variable does not appear in every term, it cannot be part of the GCF.
After factoring, the terms inside the parentheses should have no common factor.
A negative leading term is a signal to consider a negative GCF.
GCF = 1 means no GCF factoring is possible — but other methods may still apply.
After the GCF, always ask: can the remaining polynomial be factored further?
Interactive Practice — 5 Questions
What is the GCF of 12x³ and 8x²?
Which is the correct factored form of 6x² − 9x?
Factor: −5x² + 10x
Factor completely: 4x² − 36. Show all steps.
A student factors 8x³ + 12x² as 4(2x³ + 3x²). Identify the error and give the correct factored form.