Factoring by GCF

Factoring by GCF is distribution run in reverse. Distributing turns $5x(x + 4)$ into $5x^{2} + 20x$ ; factoring starts from $5x^{2} + 20x$ and pulls the shared piece back out front to get $5x(x + 4)$ . It is the first factoring move you learn, and the first move you should check for on every factoring problem afterward.

The GCF — greatest common factor — is the biggest piece that divides every term: the largest number that divides all the coefficients, times each shared variable raised to its lowest power. Find it, pull it out, and write what remains inside parentheses.

Finding the GCF

Handle numbers and variables separately. For $6x^{2} + 12x$ : the biggest number dividing both $6$ and $12$ is $6$ . Both terms contain $x$ , and the lowest power that appears is $x^{1}$ . So the GCF is $6x$ .

Why the lowest power? Because the GCF must divide every term. $x^{2}$ divides $x^{2}$ but not $x$ , so only $x^{1}$ is safe to pull from both. If a variable is missing from any term, it cannot be part of the GCF at all.

Pulling it out

Divide each term by the GCF and write the results inside parentheses. For $6x^{2} + 12x$ with GCF $6x$ : $6x^{2} \div 6x = x$ and $12x \div 6x = 2$ , so the factored form is $6x(x + 2)$ .

Every term must leave something behind. If a term is exactly the GCF, it leaves a $1$ — for example, $12x + 12 = 12(x + 1)$ , never $12(x)$ . Dropping that $1$ is the classic error here.

Check by distributing: $6x(x + 2) = 6x^{2} + 12x$ . If distributing your answer rebuilds the original polynomial exactly, the factoring is right.

Factor completely

Pull out the greatest common factor, not just any common factor. From $8x^{3} + 16x^{2}$ you could pull a $2x$ , but that leaves $2x(4x^{2} + 8x)$ , which still has a common factor inside. The full GCF is $8x^{2}$ , giving $8x^{2}(x + 2)$ — nothing left inside shares a factor, so it is factored completely.

Quick test: look inside the parentheses. If the leftover terms still share a number or a variable, you did not take the whole GCF.

Worked examples

Example 1: numbers only

Factor $20x + 8$ .

GCF of the coefficients

\gcd(20, 8) = 4

The

x

appears in only one term, so it stays inside

\text{GCF} = 4

Divide each term by

4

20x \div 4 = 5x, \quad 8 \div 4 = 2

Answer: $4(5x + 2)$

Example 2: numbers and variables

Factor $6x^{3} + 9x^{2}$ .

GCF of the coefficients

\gcd(6, 9) = 3

Lowest power of

x

in both terms

x^{2}

Divide each term by

3x^{2}

6x^{3} \div 3x^{2} = 2x, \quad 9x^{2} \div 3x^{2} = 3

Answer: $3x^{2}(2x + 3)$

Example 3: a term that is exactly the GCF

Factor $8x^{4} + 12x^{2}$ .

GCF of the coefficients

\gcd(8, 12) = 4

Lowest power of

x

x^{2}

Divide each term by

4x^{2}

8x^{4} \div 4x^{2} = 2x^{2}, \quad 12x^{2} \div 4x^{2} = 3

Check by distributing

4x^{2}(2x^{2} + 3) = 8x^{4} + 12x^{2}

Answer: $4x^{2}(2x^{2} + 3)$

Try one yourself

Common questions

What if the terms share nothing but

1

Then there is no GCF to pull out, and this method does not apply — move on to another factoring technique, like the trinomial patterns. A GCF of $1$ changes nothing.

Why do I take the lowest power of the variable?

The GCF has to divide every term evenly. The lowest power present is the most the terms all share — pulling out a higher power would leave a negative exponent behind, which means it did not actually divide that term.

What happens when a term equals the GCF exactly?

It leaves a $1$ inside the parentheses. $12x + 12$ factors to $12(x + 1)$ . Distribute to check: $12(x + 1) = 12x + 12$ , but $12(x)$ would only give $12x$ — the constant would vanish.

Why does GCF factoring come first, before other factoring?

Pulling the GCF out makes what remains smaller and simpler, and many trinomials only match the standard patterns after the GCF is gone. Checking for a GCF is step one of every factoring problem in this unit.

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Factoring by GCF

Finding the GCF

Pulling it out

Factor completely

Worked examples

Example 1: numbers only

Example 2: numbers and variables

Example 3: a term that is exactly the GCF

Try one yourself

Common questions

Keep learning

Distributing Polynomials

Factoring Trinomials (a = 1)

Factoring Trinomials (a > 1)

Factoring Binomials (Difference of Squares)