Introduction to Factoring Polynomials

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.

Reading the first equation from left to right, $8$ and $7$ are factors and $56$ is the product. Reading it from right to left instead — starting from the product and breaking it into factors — is factoring.

In an earlier chapter we factored 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 will find the greatest common factor of two numbers.

Example. Find the greatest common factor of $24$ and $36$.

Factor each number into primes, writing all variables with exponents in expanded form:

List all the factors, matching common factors in a column, and circle the factors that are shared by both numbers — here, two $2\text{s}$ and one $3$:

Bring down the common factors and multiply them:

The GCF of $24$ and $36$ is $12$. Notice that since the GCF is a factor of both numbers, $24$ and $36$ can be written as multiples of $12$:

In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.

Example. Find the greatest common factor of $5x$ and $15$.

Factor each number into primes, and circle the common factors in each column:

Bring down the common factor: $\text{GCF} = 5$. The GCF of $5x$ and $15$ is $5$.

In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

Example. Find the greatest common factor of $12x^2$ and $18x^3$.

Factor each coefficient into primes and write the variables with exponents in expanded form, then circle the common factors in each column:

Bring down the common factors and multiply:

The GCF of $12x^2$ and $18x^3$ is $6x^2$.

Example. Find the greatest common factor of $14x^3$, $8x^2$, and $10x$.

Factor each coefficient into primes and write the variables with exponents in expanded form, then circle the common factors:

All three share one factor of $2$ and one factor of $x$:

The GCF of $14x^3$, $8x^2$, and $10x$ is $2x$.

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 greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

Here, 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 we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product.

Example. Factor: $2x + 14$.

Find the GCF of $2x$ and $14$: it is $2$. Rewrite each term as a product using the GCF:

Use the Distributive Property “in reverse” to factor the expression:

Check by multiplying the factors: $2(x + 7) = 2 \cdot x + 2 \cdot 7 = 2x + 14$. ✓

Notice that in the example above we used the word factor as both a noun and a verb: $7$ is a factor of $14$ (noun), and we factor $2$ from $2x + 14$ (verb).

Example. Factor: $3a + 3$.

Find the GCF of $3a$ and $3$: it is $3$. Rewrite each term as a product using the GCF, then factor:

Check: $3(a + 1) = 3 \cdot a + 3 \cdot 1 = 3a + 3$. ✓

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: $12x - 60$.

Find the GCF of $12x$ and $60$: 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 as a product using the GCF, then factor:

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

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

Example. Factor: $3y^2 + 6y + 9$.

Find the GCF of $3y^2$, $6y$, and $9$: it is $3$. Rewrite each term as a product using the GCF, then factor:

Check: $3(y^2 + 2y + 3) = 3 \cdot y^2 + 3 \cdot 2y + 3 \cdot 3 = 3y^2 + 6y + 9$. ✓

In the next example, we factor a variable from a binomial.

Example. Factor: $6x^2 + 5x$.

Find the GCF of $6x^2$ and $5x$ and the math that goes with it: it is $x$. Rewrite each term as a product, then factor:

Check: $x(6x + 5) = x \cdot 6x + x \cdot 5 = 6x^2 + 5x$. ✓

When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!

Example. Factor: $4x^3 - 20x^2$.

Find the GCF of $4x^3$ and $20x^2$: since $4x^3 = 2 \cdot 2 \cdot x \cdot x \cdot x$ and $20x^2 = 2 \cdot 2 \cdot 5 \cdot x \cdot x$, the GCF is $4x^2$. Rewrite each term and factor:

Check: $4x^2(x - 5) = 4x^2 \cdot x - 4x^2 \cdot 5 = 4x^3 - 20x^2$. ✓

Example. Factor: $21y^2 + 35y$.

Find the GCF of $21y^2$ and $35y$: since $21y^2 = 3 \cdot 7 \cdot y \cdot y$ and $35y = 5 \cdot 7 \cdot y$, the GCF is $7y$. Rewrite each term and factor:

Example. Factor: $14x^3 + 8x^2 - 10x$.

Previously, we found the GCF of $14x^3$, $8x^2$, and $10x$ to be $2x$. Rewrite each term using the GCF, then factor:

Check: $2x(7x^2 + 4x - 5) = 2x \cdot 7x^2 + 2x \cdot 4x - 2x \cdot 5 = 14x^3 + 8x^2 - 10x$. ✓

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

Example. Factor: $-9y - 27$.

When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of $9y$ and $27$ is $9$. Since the expression $-9y - 27$ has a negative leading coefficient, we use $-9$ as the GCF. Rewrite each term using the GCF, then factor:

Check: $-9(y + 3) = -9 \cdot y + (-9) \cdot 3 = -9y - 27$. ✓

Pay close attention to the signs of the terms in the next example.

Example. Factor: $-4a^2 + 16a$.

The leading coefficient is negative, so the GCF will be negative. Ignoring signs, the GCF of $4a^2$ and $16a$ is $4a$, so we use $-4a$ as the GCF. Rewrite each term using the GCF, then factor:

Check on your own by multiplying the factors.

Key terms

factoring — splitting a product into the factors whose product it is; the reverse of multiplying. greatest common factor (GCF) — of two or more expressions, the largest expression that is a factor of all of them.

Practice

Find the greatest common factor of two or more expressions

Factor the greatest common factor from a polynomial

This section is adapted from Prealgebra 2e, Section 10.6: Introduction to Factoring Polynomials 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: recreated the factoring/multiplying diagram as typeset math and the factor-column work as inline math; omitted the Be Prepared quiz, Media links, and Self Check checklist; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block.