Greatest Common Factor and Factor by Grouping

Find the greatest common factor of two or more expressions

Earlier we multiplied factors together to get a product. Now we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

First we’ll find the GCF of two numbers.

Example. Find the GCF of $54$ and $36$.

Factor each coefficient into primes, and line up the common factors in columns. We circle the $2$, $3$, and $3$ that are shared by both numbers:

Bring down the common factors that both share — one $2$, and two $3\text{s}$ — then multiply:

The GCF of $54$ and $36$ is $18$. Notice that, because the GCF is a factor of both numbers, $54$ and $36$ can be written as multiples of $18$:

We summarize the steps we use to find the GCF below.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

Example. Find the greatest common factor of $27x^3$ and $18x^4$.

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column, then bring them down:

The shared factors are two $3\text{s}$ and three $x\text{s}$, so:

The GCF of $27x^3$ and $18x^4$ is $9x^3$.

Example. Find the GCF of $4x^2 y$ and $6xy^3$.

Factor each coefficient into primes and expand the variables. The shared factors are one $2$, one $x$, and one $y$:

The GCF of $4x^2 y$ and $6xy^3$ is $2xy$.

We can also find the greatest common factor of more than two expressions.

Example. Find the GCF of $21x^3$, $9x^2$, and $15x$.

Factor each coefficient into primes and expand the variables. Every term shares one $3$ and one $x$:

The GCF of $21x^3$, $9x^2$, and $15x$ is $3x$.

Factor the greatest common factor from a polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, $12$ as $2 \cdot 6$ or $3 \cdot 4$), in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial using the Distributive Property:

Now we will start with a product, like $2x + 14$, and end with its factors, $2(x + 7)$. To do this we apply the Distributive Property “in reverse.”

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

Example. Factor $4x + 12$.

First find the GCF of all the terms. Since $4x = 2 \cdot 2 \cdot x$ and $12 = 2 \cdot 2 \cdot 3$, the GCF is $2 \cdot 2 = 4$. Rewrite each term as a product using the GCF, then use the “reverse” Distributive Property to factor:

Check by multiplying: $4(x + 3) = 4 \cdot x + 4 \cdot 3 = 4x + 12$. ✓

We use “factor” as both a noun and a verb. As a noun, $7$ is a factor of $14$. As a verb, we factor $3$ from $3a + 3$.

Example. Factor $5a + 5$.

The GCF of $5a$ and $5$ is $5$. Rewrite each term as a product using the GCF, then factor:

Check by multiplying: $5(a + 1) = 5 \cdot a + 5 \cdot 1 = 5a + 5$. ✓

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Example. Factor $12x - 60$.

Since $12x = 2 \cdot 2 \cdot 3 \cdot x$ and $60 = 2 \cdot 2 \cdot 3 \cdot 5$, the GCF is $2 \cdot 2 \cdot 3 = 12$. Rewrite each term and factor:

Check by multiplying: $12(x - 5) = 12 \cdot x - 12 \cdot 5 = 12x - 60$. ✓

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Example. Factor $4y^2 + 24y + 28$.

Find the GCF of all three terms. Since $4y^2 = 2 \cdot 2 \cdot y \cdot y$, $24y = 2 \cdot 2 \cdot 2 \cdot 3 \cdot y$, and $28 = 2 \cdot 2 \cdot 7$, the GCF is $2 \cdot 2 = 4$. Rewrite each term as a product using the GCF, then factor:

Check by multiplying: $4(y^2 + 6y + 7) = 4y^2 + 24y + 28$. ✓

Example. Factor $5x^3 - 25x^2$.

The GCF of $5x^3$ and $25x^2$ is $5x^2$. Rewrite each term and factor:

Check by multiplying: $5x^2(x - 5) = 5x^2 \cdot x - 5x^2 \cdot 5 = 5x^3 - 25x^2$. ✓

Example. Factor $21x^3 - 9x^2 + 15x$.

In a previous example we found the GCF of $21x^3$, $9x^2$, and $15x$ to be $3x$. Rewrite each term using the GCF, then factor:

Check by multiplying: $3x(7x^2 - 3x + 5) = 21x^3 - 9x^2 + 15x$. ✓

Example. Factor $8m^3 - 12m^2 n + 20mn^2$.

Since $8m^3 = 2 \cdot 2 \cdot 2 \cdot m \cdot m \cdot m$, $12m^2 n = 2 \cdot 2 \cdot 3 \cdot m \cdot m \cdot n$, and $20mn^2 = 2 \cdot 2 \cdot 5 \cdot m \cdot n \cdot n$, the GCF is $2 \cdot 2 \cdot m = 4m$. Rewrite each term and factor:

Check by multiplying: $4m(2m^2 - 3mn + 5n^2) = 8m^3 - 12m^2 n + 20mn^2$. ✓

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Example. Factor $-8y - 24$.

When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, the GCF of $8y$ and $24$ is $8$. Since the expression $-8y - 24$ has a negative leading coefficient, we use $-8$ as the GCF. Rewrite each term and factor:

Check by multiplying: $-8(y + 3) = -8 \cdot y + (-8) \cdot 3 = -8y - 24$. ✓

Example. Factor $-6a^2 + 36a$.

The leading coefficient is negative, so the GCF is negative, $-6a$. Rewrite each term using the GCF, then factor:

Check by multiplying: $-6a(a - 6) = -6a \cdot a + (-6a)(-6) = -6a^2 + 36a$. ✓

The greatest common factor doesn’t have to be a monomial — it can be a binomial too.

Example. Factor $5q(q + 7) - 6(q + 7)$.

The GCF here is the binomial $q + 7$. Factor it out from each term:

Check on your own by multiplying.

Factor by grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)

Example. Factor $xy + 3y + 2x + 6$.

There is no greatest common factor of all four terms, so let’s separate the first two terms from the second two terms. Factor the GCF from each group: from $xy + 3y$ we factor $y$, and from $2x + 6$ we factor $2$:

Notice that each term now has a common factor of $(x + 3)$, so we factor it out. Check by multiplying: $(x + 3)(y + 2) = xy + 2x + 3y + 6$. ✓

Example. Factor $x^2 + 3x - 2x - 6$.

There is no GCF in all four terms. Separate the polynomial into two parts: $x^2 + 3x$ and $-2x - 6$. Factor the GCF from both parts, being careful with the signs when factoring the GCF from the last two terms:

Check on your own by multiplying.

Key terms

factoring — splitting a product into the factors that multiply to give it. greatest common factor (GCF) — the largest expression that is a factor of each of two or more given expressions. factor by grouping — a method for factoring a four-term polynomial by grouping terms with common factors, factoring the GCF from each group, and then factoring out the common binomial.

This section is adapted from Elementary Algebra 2e, Section 7.1: Greatest Common Factor and Factor by Grouping by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the GCF prime-factorization tables and worked factoring examples as prose with display equality chains, and stated each GCF as a bring-down product; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.