Lesson 4: Factoring Polynomials

Property

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

HOW TO: Find the Greatest Common Factor (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

• Find the GCF of $42$ and $70$. We factor each number: $42 = 2 \cdot 3 \cdot 7$ and $70 = 2 \cdot 5 \cdot 7$. The common factors are $2$ and $7$. So, the GCF is $2 \cdot 7 = 14$.
• Find the GCF of $15a^2$ and $25a^3$. We factor each term: $15a^2 = 3 \cdot 5 \cdot a \cdot a$ and $25a^3 = 5 \cdot 5 \cdot a \cdot a \cdot a$. The common factors are $5, a, a$. The GCF is $5a^2$.
• Find the GCF of $12x^2y$ and $18xy^2$. We factor each term: $12x^2y = 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot y$ and $18xy^2 = 2 \cdot 3 \cdot 3 \cdot x \cdot y \cdot y$. The common factors are $2, 3, x, y$. The GCF is $6xy$.

Property

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

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

HOW TO: Factor the greatest common factor from a polynomial.
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 "reverse" Distributive Property to factor the expression.
Step 4. Check by multiplying the factors.

Examples

• Factor $7y + 21$. The GCF of $7y$ and $21$ is $7$. We rewrite the expression as $7 \cdot y + 7 \cdot 3$. Using the reverse distributive property, we get $7(y+3)$.
• Factor $6a^3 - 18a^2 + 12a$. The GCF of all terms is $6a$. We rewrite this as $6a \cdot a^2 - 6a \cdot 3a + 6a \cdot 2$. Factoring out the GCF gives $6a(a^2 - 3a + 2)$.
• Factor $-5x - 20$. When the leading term is negative, we factor out a negative GCF. The GCF is $-5$. This gives $-5(x) + (-5)(4)$, which factors to $-5(x+4)$.