Factoring Out the Greatest Common Factor

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Factoring out the GCF means finding the greatest common factor of every term in a polynomial and writing it in front of parentheses. It is the distributive property run backward. For example, 6x^2 + 9x = 3x(2x + 3), because 3x divides both terms. Always look for a GCF first when you factor.

Before this lesson: You should know how to find the greatest common factor of numbers, the distributive property, and the exponent rules.

What Factoring Means

To factor means to write something as a product. You already factor numbers: 12 = 4 \cdot 3. Now you will factor polynomials.

When you distribute, you multiply: 3x(2x + 3) = 6x^2 + 9x. When you factor, you go the other way. You start with 6x^2 + 9x and find the 3x(2x + 3) it came from.

A rectangle split into two parts with areas 6x² and 9x. The shared height is 3x, the GCF. The widths of the parts are 2x and 3. Beside it: multiplying 3x(2x + 3) gives 6x² + 9x, and factoring 6x² + 9x gives 3x(2x + 3).
The two areas share a side of 3x. Factoring finds that side and the other side, 2x + 3.

Think of the polynomial as the area of a rectangle. The GCF is one side. The part in parentheses is the other side.

How to Factor Out the GCF

The greatest common factor (GCF) of a polynomial is the largest monomial that divides every term. It has two parts.

Part of the GCFHow to find itExample: 12x^3 and 18x^2
NumberThe GCF of the coefficients6
Each variableThe smallest exponent that appears in every termx^2
Whole GCFMultiply the two parts6x^2

So 12x^3 + 18x^2 = 6x^2(2x + 3). To find each term inside, divide by the GCF: 12x^3 \div 6x^2 = 2x and 18x^2 \div 6x^2 = 3.

Steps

  1. Find the GCF of the coefficients.
  2. For each variable that is in every term, take the smallest exponent.
  3. Write the GCF in front of a set of parentheses.
  4. Divide each term by the GCF. Write the answers inside the parentheses.
  5. Check by distributing. You should get the original polynomial.

Worked Examples

Example 1: A number GCF

Factor 8x + 12.

The GCF of 8 and 12 is 4. Only the first term has an x, so x is not part of the GCF.

\begin{aligned} 8x + 12 &= 4 \cdot 2x + 4 \cdot 3 \\ &= 4(2x + 3) \end{aligned}

Check: 4(2x + 3) = 8x + 12.

Example 2: A variable in the GCF

Factor 10x^3 - 15x^2.

The GCF of 10 and 15 is 5. The smallest power of x is x^2. So the GCF is 5x^2.

\begin{aligned} 10x^3 - 15x^2 &= 5x^2 \cdot 2x - 5x^2 \cdot 3 \\ &= 5x^2(2x - 3) \end{aligned}

Example 3: Two variables and three terms

Factor 12a^3b^2 + 18a^2b^3 - 6a^2b.

The GCF of 12, 18 and 6 is 6. The smallest powers are a^2 and b. So the GCF is 6a^2b.

\begin{aligned} &= 6a^2b \cdot 2ab + 6a^2b \cdot 3b^2 - 6a^2b \cdot 1 \\ &= 6a^2b(2ab + 3b^2 - 1) \end{aligned}

The last term divided by the GCF is 1. Don’t leave it out. The parentheses must still have three terms.

Example 4: A negative leading coefficient

Factor -4x^2 + 20x.

When the first term is negative, factor out a negative GCF. Then the first term inside is positive. The GCF is -4x.

\begin{aligned} -4x^2 + 20x &= -4x \cdot x + (-4x)(-5) \\ &= -4x(x - 5) \end{aligned}

Dividing by a negative changes every sign: 20x \div (-4x) = -5.

Example 5: Word problem

A rectangular patio has an area of 12x^2 + 8x square feet. Its width is 4x feet. Find its length. Then find the length and area when x = 3.

Area is width times length. So factor out the width, 4x, which is also the GCF.

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

The length is 3x + 2 feet. When x = 3, the width is 12 feet and the length is 11 feet. The area is 12 \cdot 11 = 132 square feet, and 12(9) + 8(3) = 132 checks it.

Common Mistakes

MistakeExample of the errorFix
Leaving out the 16x^2 + 3x = 3x(2x)3x \div 3x = 1, so 3x(2x + 1).
Not using the greatest factor12x + 18 = 2(6x + 9)6 and 9 still share 3. Use 6: 6(2x + 3).
Using the largest exponentx^4 + x^2 = x^4(1 + x^2)Use the smallest exponent: x^2(x^2 + 1).
Sign errors with a negative GCF-3x - 6 = -3(x - 6)-6 \div (-3) = 2, so -3(x + 2).
Dividing only some terms5x^2 + 10x - 15 = 5(x^2 + 10x - 15)Divide every term: 5(x^2 + 2x - 3).

More Ways the GCF Shows Up

When the GCF is a binomial

Sometimes every term shares a whole binomial. Treat it like one big factor.

3x(x + 2) + 5(x + 2) = (x + 2)(3x + 5)

Both terms have the factor (x + 2). Pull it out, and what is left is 3x + 5. You will use this idea again in factoring by grouping.

Factor out the GCF first

Taking out the GCF first makes the rest easier to factor. For example, 2x^2 + 10x + 12 = 2(x^2 + 5x + 6). The trinomial inside has smaller numbers, so it is easier to factor in the next lesson, Factoring Trinomials.

When the GCF is 1

Some polynomials, like 3x + 5 or x^2 + 7, have no common factor except 1. You can’t pull out a GCF. Some of them may still factor in other ways.

Factoring also undoes the special products of binomials and other polynomial products, as you will see in this unit.

Practice Problems

  1. Factor 7x + 21.
  2. Factor 9y^2 - 6y.
  3. Factor 4x^3 + 8x^2 - 12x.
  4. Factor 15a^2b - 25ab^2.
  5. Factor 2x^4 + 6x^3 + 2x^2.
  6. Factor -6x^2 - 18x.
  7. Factor 14m^3n^2 + 21m^2n^2 - 7mn^2.
  8. Factor x(x - 4) + 9(x - 4).
  9. Find the GCF of 24x^3, 36x^2 and 60x.
  10. A rectangle has an area of 18x^2 + 27x square meters and a width of 9x meters. Find its length. Then find the area when x = 2.
Show answers
  1. 7(x + 3)
  2. 3y(3y - 2). Check: 9y^2 - 6y.
  3. 4x(x^2 + 2x - 3)
  4. 5ab(3a - 5b)
  5. 2x^2(x^2 + 3x + 1). The last term inside is 1.
  6. -6x(x + 3). The first term is negative, so factor out -6x.
  7. 7mn^2(2m^2 + 3m - 1)
  8. (x - 4)(x + 9). The common factor is the binomial x - 4.
  9. 12x. The GCF of 24, 36 and 60 is 12, and the smallest power of x is x^1.
  10. 18x^2 + 27x = 9x(2x + 3), so the length is 2x + 3 meters. When x = 2: 18 \cdot 7 = 126 square meters.

FAQ

What does it mean to factor out the GCF?

It means you divide every term by their greatest common factor and write the GCF in front of parentheses. For example, 5x + 10 = 5(x + 2).

How do you find the GCF of a polynomial?

Find the GCF of the coefficients. Then, for each variable in every term, use the smallest exponent. Multiply these together.

Should I factor out a negative?

Usually yes, when the first term is negative. Factoring out a negative GCF leaves a positive first term, like -2x^2 + 8x = -2x(x - 4). Remember to change every sign inside.

What if there is no common factor?

Then the GCF is 1, and you can’t pull anything out. The polynomial may still factor another way, such as a trinomial like x^2 + 5x + 6.

Summary

  • Factoring writes a polynomial as a product. It undoes distributing.
  • The GCF uses the GCF of the coefficients and the smallest power of each shared variable.
  • Divide every term by the GCF and write the results in parentheses.
  • If a term equals the GCF, a 1 is left in its place.
  • Check by distributing. Factor out the GCF first in every factoring problem.

Previous lesson: Special Products of Binomials

Next lesson: Factoring Trinomials

Back to the Algebra 1 Index

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