Factoring Out the GCF

Factoring is the reverse of multiplying. When you multiply $3(x + 5)$, you get $3x + 15$. Factoring starts with $3x + 15$ and works backward to find $3(x + 5)$. The first step in any factoring problem is always the same: pull out the greatest common factor (GCF).

The GCF of two or more terms is the largest expression — number, variable, or both — that divides evenly into every term. Mastering this skill is essential because it simplifies expressions, makes equations easier to solve, and is the foundation for every other factoring technique you will learn.

Finding the GCF of Monomials

Before you can factor the GCF out of a polynomial, you need to know how to find the GCF of individual terms.

Step 1 — Find the GCF of the numerical coefficients. List the factors of each coefficient and pick the largest one they share.

Step 2 — Find the GCF of the variables. For each variable that appears in every term, take the lowest exponent.

Step 3 — Multiply the results from Steps 1 and 2.

Example 1: Find the GCF of $12x^3$ and $18x^2$

Coefficients: The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The GCF of 12 and 18 is 6.

Variables: Both terms have $x$. The lowest exponent is 2. So the variable part is $x^2$.

$\text{GCF} = 6x^2$

Example 2: Find the GCF of $20a^4b^2$, $30a^2b^5$, and $10a^3b$

Coefficients: GCF of 20, 30, and 10 is 10.

Variables: The lowest power of $a$ is $a^2$. The lowest power of $b$ is $b^1 = b$.

$\text{GCF} = 10a^2b$

Factoring the GCF from a Polynomial

Once you identify the GCF, divide each term by it and write the result in parentheses.

$\text{polynomial} = \text{GCF} \times \left(\frac{\text{term}_1}{\text{GCF}} + \frac{\text{term}_2}{\text{GCF}} + \cdots \right)$

Example 3: Factor $6x^2 + 9x$

Step 1 — Find the GCF: The GCF of 6 and 9 is 3. Both terms have at least $x^1$. So the GCF is $3x$.

Step 2 — Divide each term by $3x$:

$\frac{6x^2}{3x} = 2x \qquad \frac{9x}{3x} = 3$

Step 3 — Write the factored form:

$6x^2 + 9x = 3x(2x + 3)$

Check by distributing: $3x(2x + 3) = 6x^2 + 9x$ . Correct.

Example 4: Factor $15x^3 - 25x^2 + 10x$

Step 1 — Find the GCF: The GCF of 15, 25, and 10 is 5. The lowest power of $x$ is $x^1$. So the GCF is $5x$.

Step 2 — Divide each term:

$\frac{15x^3}{5x} = 3x^2 \qquad \frac{-25x^2}{5x} = -5x \qquad \frac{10x}{5x} = 2$

Step 3 — Write the factored form:

$15x^3 - 25x^2 + 10x = 5x(3x^2 - 5x + 2)$

Check: $5x(3x^2 - 5x + 2) = 15x^3 - 25x^2 + 10x$ . Correct.

Example 5: Factor $8a^3b^2 + 12a^2b^3 - 4a^2b^2$

Step 1 — Find the GCF: GCF of 8, 12, and 4 is 4. Lowest power of $a$ is $a^2$. Lowest power of $b$ is $b^2$. So the GCF is $4a^2b^2$.

Step 2 — Divide each term:

$\frac{8a^3b^2}{4a^2b^2} = 2a \qquad \frac{12a^2b^3}{4a^2b^2} = 3b \qquad \frac{-4a^2b^2}{4a^2b^2} = -1$

Step 3 — Write the factored form:

$8a^3b^2 + 12a^2b^3 - 4a^2b^2 = 4a^2b^2(2a + 3b - 1)$

Notice that the last term became $-1$, not 0. When a term is exactly equal to the GCF, it leaves behind a 1 (or $-1$).

Factoring Out a Negative Leading Coefficient

When the leading term (the first term) has a negative coefficient, it is often helpful to factor out the negative sign along with the GCF. This is especially useful before applying other factoring techniques later.

Example 6: Factor $-12x^2 + 18x$

Step 1 — The GCF of 12 and 18 is 6, and both terms have $x$. Factor out $-6x$ (including the negative):

$\frac{-12x^2}{-6x} = 2x \qquad \frac{18x}{-6x} = -3$

Step 2 — Write the factored form:

$-12x^2 + 18x = -6x(2x - 3)$

Check: $-6x(2x - 3) = -12x^2 + 18x$ . Correct.

Notice how factoring out the negative flipped the signs inside the parentheses. You could also factor out $6x$ to get $6x(-2x + 3)$, but the convention is to make the leading term inside the parentheses positive when possible.

When the GCF Is 1

Sometimes the terms share no common factor other than 1. In that case, you cannot factor out a GCF and should move on to other techniques (trinomial factoring, difference of squares, grouping). Recognizing when the GCF is 1 saves time.

Example 7: Consider $x^2 + 5x + 6$

The coefficients are 1, 5, and 6 — their GCF is 1. There is no common variable factor (the constant 6 has no $x$). The GCF is 1, so there is nothing to factor out. You would proceed with trinomial factoring instead.

Real-World Application: Retail — Simplifying a Revenue Formula

A small retail store sells t-shirts and calculates monthly revenue with the expression:

$R = 15n + 15(0.08n)$

where $n$ is the number of shirts sold and the second term accounts for an 8% sales tax collected. The store owner wants to simplify this formula.

Factor out 15n:

$R = 15n + 1.2n = n(15 + 1.2) = 16.2n$

Wait — let us factor the original expression more carefully. Both terms share the factor $15n$:

$R = 15n(1 + 0.08) = 15n(1.08)$

So the total revenue per shirt (including tax) is $15 times 1.08, giving $16.20 per shirt. Factoring reveals the structure: base price times the tax multiplier. This is the same technique — pulling out the common factor to see what is really going on.

Real-World Application: Carpentry — Calculating Material for a Deck Frame

A carpenter needs lumber for a rectangular deck frame. The perimeter formula gives the total linear feet of lumber:

$P = 2L + 2W$

Factor out the GCF of 2:

$P = 2(L + W)$

If the deck is $L = 16$ feet by $W = 12$ feet:

$P = 2(16 + 12) = 2(28) = 56 \text{ feet}$

The factored form $2(L + W)$ is faster to compute mentally and shows the structure: the perimeter is always twice the sum of length and width. When ordering lumber with a 10% waste allowance, the carpenter computes $P \times 1.10 = 56 \times 1.10 = 61.6$ feet, rounding up to 62 linear feet.

Common Mistakes to Avoid

1. Not factoring completely. If you factor $12x^2 + 6x$ as $2(6x^2 + 3x)$, you have not finished — $6x^2 + 3x$ still has a common factor of $3x$. The correct answer is $6x(2x + 1)$.
2. Dropping a term. When the GCF equals one of the terms, the quotient is 1, not 0. For example, factoring $5x + 5$ gives $5(x + 1)$, not $5(x)$.
3. Sign errors when factoring out a negative. Double-check every sign inside the parentheses by distributing back.
4. Forgetting to check your answer. Always multiply the factored form back out to verify it matches the original polynomial.
5. Overlooking variable factors. The GCF of $4x^3 + 6x$ is $2x$, not just 2. Always check both coefficients and variables.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Factoring out the GCF is always the first step in any factoring problem — check for it before trying any other technique
• To find the GCF: take the GCF of the coefficients, then the lowest exponent of each shared variable
• Divide every term by the GCF and write the results inside parentheses
• When the leading coefficient is negative, factor out the negative along with the GCF
• If the GCF is 1, move on to other factoring methods
• Always check your work by multiplying the factored form back out

Return to Algebra for more topics in this section.