Lesson 6.1: Greatest Common Factor and Factor by Grouping

New Concept

Factoring is the reverse of multiplication. In this lesson, you'll learn to find the greatest common factor (GCF) to simplify polynomials and use the "factor by grouping" technique for expressions with four terms.

What’s next

Next, you'll master finding the GCF with interactive examples. Then, you'll tackle factoring by grouping on our practice cards.

Property

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

To find the GCF of two expressions:
Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
Step 3. Bring down the common factors that all expressions share.
Step 4. Multiply the factors.

Examples

• To find the GCF of $18a^3$ and $24a^2$, we factor each term. $18a^3 = 2 \cdot 3 \cdot 3 \cdot a \cdot a \cdot a$ and $24a^2 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot a \cdot a$. The common factors are $2 \cdot 3 \cdot a \cdot a$, so the GCF is $6a^2$.

Property

We use the Distributive Property in reverse to factor a polynomial. Find the GCF of all the terms and write the polynomial as a product.

Distributive Property:
If $a$, $b$, and $c$ are real numbers, then $a(b + c) = ab + ac$ and $ab + ac = a(b + c)$. The form on the right is used to factor.

To factor the GCF from a polynomial:
Step 1. Find the GCF of all terms.
Step 2. Rewrite each term as a product using the GCF.
Step 3. Use the “reverse” Distributive Property to factor the expression.
Step 4. Check by multiplying the factors.

Property

When the leading coefficient is negative, we factor the negative out as part of the GCF. This changes the signs of the terms inside the parentheses.

For example, to factor $-4a^3 + 36a^2 - 8a$, the GCF is taken as $-4a$. We rewrite each term using the GCF:
$-4a^3 = -4a \cdot a^2$
$36a^2 = -4a \cdot (-9a)$
$-8a = -4a \cdot 2$
The factored expression is $-4a(a^2 - 9a + 2)$.

Examples

• Factor $-7x - 21$. The GCF is $-7$. Factoring it out gives $-7(x + 3)$. Notice the sign inside the parenthesis flipped.

Property

When a polynomial has four terms and no GCF for all terms, you can sometimes factor by grouping.

Step 1. Group terms with common factors, usually into two pairs.
Step 2. Factor out the common factor in each group.
Step 3. Factor the common binomial factor from the resulting expression.
Step 4. Check by multiplying the factors.

Examples

• Factor $xy + 8y + 4x + 32$. Group as $(xy + 8y) + (4x + 32)$. Factor each group: $y(x + 8) + 4(x + 8)$. The common binomial is $(x + 8)$, so the final answer is $(x + 8)(y + 4)$.