Greatest Common Factor and Factor by Grouping

By the end of this section, you will be able to: find the greatest common factor of two or more expressions, factor the greatest common factor from a polynomial, 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 reverse this process: we start with a product and break it down into its factors. Splitting a product into factors is called factoring. For example, $8 \cdot 7 = 56$ and $2x(x+3)=2x^2+6x$ show multiplication; reversing either process finds the factors of the product.

We have already factored numbers to find the least common multiple of two or more numbers. Now we factor expressions and find their greatest common factor. The method is similar to the method used to find the LCM.

Greatest common factor. The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

Find the greatest common factor (GCF) of two or more expressions.

Factor each coefficient into primes. Write all variables with exponents in expanded form.
List all factors, matching common factors in a column. In each column, identify the common factors.
Bring down the common factors that all expressions share.
Multiply the factors.

Example. Find the greatest common factor of $21x^3$ , $9x^2$ , and $15x$ .

Factor each coefficient into primes and expand each power. The factors common to all three expressions are $3$ and $x$ :

\begin{array}{rcl} 21x^3 &=& 3 \cdot 7 \cdot x \cdot x \cdot x \\[4pt] 9x^2 &=& 3 \cdot 3 \cdot x \cdot x \\[4pt] 15x &=& 3 \cdot 5 \cdot x \end{array}

Bring down and multiply the common factors:

\text{GCF}=3 \cdot x=3x.

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

Find the greatest common factor of $25m^4$ , $35m^3$ , and $20m^2$ .

Find the greatest common factor of $14x^3$ , $70x^2$ , and $105x$ .

Factor the greatest common factor from a polynomial

It is sometimes useful to represent a number as a product of factors. For example, $12=2 \cdot 6=3 \cdot 4$ . In algebra, it is also useful to represent a polynomial in factored form. We can start with the polynomial $3x^2+15x$ and end with its factors, $3x(x+5)$ , by applying the Distributive Property in reverse.

Distributive Property. If $a$ , $b$ , and $c$ are real numbers, then

a(b+c)=ab+ac \quad\text{and}\quad ab+ac=a(b+c).

The form on the left is used to multiply. The form on the right is used to factor.

To factor a polynomial, find the GCF of all its terms and write the polynomial as a product.

Example. Use the Distributive Property to factor $8m^3-12m^2n+20mn^2$ .

First find the GCF. Factoring the coefficients and expanding the variables shows that every term contains $4m$ :

\begin{array}{rcl} 8m^3 &=& 2 \cdot 2 \cdot 2 \cdot m \cdot m \cdot m \\[4pt] 12m^2n &=& 2 \cdot 2 \cdot 3 \cdot m \cdot m \cdot n \\[4pt] 20mn^2 &=& 2 \cdot 2 \cdot 5 \cdot m \cdot n \cdot n \end{array}

Rewrite each term as a product using $4m$ , and then use the Distributive Property in reverse:

\begin{array}{lrcl} \text{Rewrite each term.} & 8m^3-12m^2n+20mn^2 &=& 4m \cdot 2m^2-4m \cdot 3mn+4m \cdot 5n^2 \\[4pt] \text{Factor the GCF.} &&=& 4m(2m^2-3mn+5n^2) \end{array}

Multiplying the factors returns the original polynomial, so the factorization checks.

Factor $9xy^2+6x^2y^2+21y^3$ by taking out the greatest common factor.

Factor $3p^3-6p^2q+9pq^3$ by taking out the greatest common factor.

Factor the greatest common factor from a polynomial.

Find the GCF of all the terms of the polynomial.
Rewrite each term as a product using the GCF.
Use the reverse Distributive Property to factor the expression.
Check by multiplying the factors.

Factor as a noun and a verb

We use factor as both a noun and a verb. In the statement “ $7$ is a factor of $14$ ,” factor is a noun. In the instruction “factor $3$ from $3a+3$ ,” it is a verb.

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

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

\begin{array}{lrcl} \text{Rewrite each term.} & 5x^3-25x^2 &=& 5x^2 \cdot x-5x^2 \cdot 5 \\[4pt] \text{Factor the GCF.} &&=& 5x^2(x-5) \end{array}

Check: $5x^2(x-5)=5x^3-25x^2$ .

Factor $2x^3+12x^2$ by taking out the greatest common factor.

Factor $6y^3-15y^2$ by taking out the greatest common factor.

Example. Factor $8x^3y-10x^2y^2+12xy^3$ .

The GCF of the three terms is $2xy$ . Rewrite each term using the GCF and factor:

\begin{array}{lrcl} \text{Rewrite each term.} & 8x^3y-10x^2y^2+12xy^3 &=& 2xy \cdot 4x^2-2xy \cdot 5xy+2xy \cdot 6y^2 \\[4pt] \text{Factor the GCF.} &&=& 2xy(4x^2-5xy+6y^2) \end{array}

Check: multiplying $2xy$ through the trinomial gives the original polynomial.

Factor $15x^3y-3x^2y^2+6xy^3$ by taking out the greatest common factor.

Factor $8a^3b+2a^2b^2-6ab^3$ by taking out the greatest common factor.

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

Example. Factor $-4a^3+36a^2-8a$ .

Because the leading coefficient is negative, use the negative GCF $-4a$ :

\begin{array}{lrcl} \text{Rewrite each term.} & -4a^3+36a^2-8a &=& -4a \cdot a^2+(-4a)(-9a)+(-4a) \cdot 2 \\[4pt] \text{Factor the GCF.} &&=& -4a(a^2-9a+2) \end{array}

Check: $-4a(a^2-9a+2)=-4a^3+36a^2-8a$ .

Factor $-4b^3+16b^2-8b$ by taking out a negative greatest common factor.

Factor $-7a^3+21a^2-14a$ by taking out a negative greatest common factor.

So far the greatest common factors have been monomials. A GCF can also be a binomial.

Example. Factor $3y(y+7)-4(y+7)$ .

The binomial $y+7$ is the common factor:

3y(y+7)-4(y+7)=(y+7)(3y-4).

Check by multiplying the factors.

Factor $4m(m+3)-7(m+3)$ by taking out the common binomial factor.

Factor $8n(n-4)+5(n-4)$ by taking out the common binomial factor.

Factor by grouping

Sometimes there is no common factor of all the terms of a polynomial. When there are four terms, separate the polynomial into two groups of two terms. Then look for the GCF in each group. If the polynomial can be factored, a common factor emerges from the two groups. Not all polynomials can be factored; just as some numbers are prime, some polynomials are prime.

Example. Factor by grouping: $xy+3y+2x+6$ .

There is no GCF of all four terms. Group the first two terms and the last two terms, then factor the GCF from each group:

\begin{array}{lrcl} \text{Group the terms.} &&& (xy+3y)+(2x+6) \\[4pt] \text{Factor each group.} &&& y(x+3)+2(x+3) \\[4pt] \text{Factor the common binomial.} &&& (x+3)(y+2) \end{array}

Check: $(x+3)(y+2)=xy+2x+3y+6$ , which is the original polynomial with its middle terms reordered.

Factor by grouping: $xy+8y+3x+24$ .

Factor by grouping: $ab+7b+8a+56$ .

Factor by grouping.

Group terms with common factors.
Factor out the common factor in each group.
Factor the common factor from the expression.
Check by multiplying the factors.

Example. Factor each polynomial by grouping.

(a) $x^2+3x-2x-6$

There is no GCF in all four terms. Separate the polynomial into two groups. Be careful with the signs when factoring the GCF from the last group:

\begin{array}{lrcl} \text{Group the terms.} &&& (x^2+3x)+(-2x-6) \\[4pt] \text{Factor each group.} &&& x(x+3)-2(x+3) \\[4pt] \text{Factor the common binomial.} &&& (x+3)(x-2) \end{array}

(b) $6x^2-3x-4x+2$

Again, group the terms, factor each group, and then factor out the common binomial:

\begin{array}{lrcl} \text{Group the terms.} &&& (6x^2-3x)+(-4x+2) \\[4pt] \text{Factor each group.} &&& 3x(2x-1)-2(2x-1) \\[4pt] \text{Factor the common binomial.} &&& (2x-1)(3x-2) \end{array}

Check both results by multiplying their factors.

Factor by grouping: $x^2+2x-5x-10$ .

Factor by grouping: $20x^2-16x-15x+12$ .

Factor by grouping: $y^2+4y-7y-28$ .

Key terms

factoring — splitting a product into its factors. greatest common factor (GCF) — the largest expression that is a factor of every expression in a given collection. factored form — an expression written as a product of factors. factor by grouping — factor a four-term polynomial by grouping terms, factoring each group, and then factoring the common binomial.

This section is adapted from Intermediate Algebra 2e, Section 6.1: Greatest Common Factor and Factor by Grouping by Lynn Marecek and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: reformatted the worked-example tables as accessible aligned math, omitted the Be Prepared quiz, media links, and end-of-section exercises, and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

Factoring Factor Trinomials