Lesson 4: Solving Polynomial Equations in Factored Form

Property

Zero Product Property
If $a \cdot b = 0$, then either $a = 0$ or $b = 0$ or both.

How To Use the Zero Product Property

1. Set each factor equal to zero.
2. Solve the linear equations.
3. Check.

Examples

• To solve $(x - 5)(3x + 2) = 0$, set each factor to zero. $x - 5 = 0$ gives $x = 5$, and $3x + 2 = 0$ gives $x = -\frac{2}{3}$.

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 $6xy$.