Lesson 6: Introduction to Factoring Polynomials

New Concept

Factoring is the reverse of multiplying. In this lesson, you'll learn to find the greatest common factor (GCF) of a polynomial's terms and use the Distributive Property in reverse to express the polynomial as a product of its factors.

What’s next

Now that you have the main idea, you'll dive into interactive examples for finding the GCF and factoring polynomials. Then, test your skills with practice cards.

Property

Splitting a product into factors is called factoring. The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

Examples

• To find the GCF of 24 and 36, we factor each number: $24 = 2 \cdot 2 \cdot 2 \cdot 3$ and $36 = 2 \cdot 2 \cdot 3 \cdot 3$. The common factors are $2 \cdot 2 \cdot 3$, so the GCF is 12.

• To find the GCF of $5x$ and 15, we factor each term: $5x = 5 \cdot x$ and $15 = 3 \cdot 5$. The only common factor is 5.

Property

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 40 and 56, factor them: $40 = 2 \cdot 2 \cdot 2 \cdot 5$ and $56 = 2 \cdot 2 \cdot 2 \cdot 7$. The common factors are $2 \cdot 2 \cdot 2$, so the GCF is 8.

• To find the GCF of $9x$ and $15x^2$, factor them: $9x = 3 \cdot 3 \cdot x$ and $15x^2 = 3 \cdot 5 \cdot x \cdot x$. The common factors are $3 \cdot x$, so the GCF is $3x$.

Property

If $a, b, c$ are real numbers, then $ab + ac = a(b + c)$.
Step 1. Find the GCF of all the terms of the polynomial.
Step 2. Rewrite each term as a product using the GCF.
Step 3. Use the Distributive Property 'in reverse' to factor the expression.
Step 4. Check by multiplying the factors.

Examples

• To factor $3a + 3$, the GCF is 3. Rewrite the expression as $3 \cdot a + 3 \cdot 1$. Using the distributive property in reverse, we get $3(a + 1)$.

• To factor $6x^2 + 5x$, the GCF is $x$. Rewrite the expression as $x \cdot 6x + x \cdot 5$. Factoring out the GCF gives $x(6x + 5)$.

Property

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

Examples

• To factor $-9y - 27$, the GCF of $9y$ and $27$ is 9. Since the leading term is negative, we use $-9$ as the GCF, which gives $-9(y + 3)$.

• To factor $-4a^2 + 16a$, the GCF is $4a$. We use $-4a$ as the GCF, which results in $-4a(a - 4)$. Notice the sign of the second term is flipped.