Lesson 6.5: Polynomial Equations

New Concept

This lesson connects factoring to solving polynomial equations. You'll learn to use the Zero Product Property—if $a \cdot b = 0$, then $a=0$ or $b=0$—to find solutions for quadratic and higher-degree equations.

What’s next

Next, you'll tackle interactive examples and practice cards on solving quadratic equations. Then, you'll apply these skills to challenge problems that model real-world scenarios.

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

Quadratic Equation
An equation of the form $ax^2 + bx + c = 0$ is called a quadratic equation. $a, b$, and $c$ are real numbers and $a \neq 0$.

How To Solve a Quadratic Equation by Factoring

1. Write the quadratic equation in standard form, $ax^2 + bx + c = 0$.
2. Factor the quadratic expression.
3. Use the Zero Product Property.
4. Solve the linear equations.
5. Check. Substitute each solution separately into the original equation.

Examples

• To solve $x^2 - 3x = 10$, first write it in standard form: $x^2 - 3x - 10 = 0$. Factoring gives $(x - 5)(x + 2) = 0$. The solutions are $x = 5$ and $x = -2$.

Property

Zero of a Function
For any function $f$, if $f(x) = 0$, then $x$ is a zero of the function.

When we find the values of $x$ for which $f(x) = 0$, we are finding the zeros of the function. When $f(x) = 0$, the point $(x, 0)$ is a point on the graph, which is an $x$-intercept.

Examples

• For the function $f(x) = x^2 - x - 8$, find $x$ when $f(x) = 4$. Set $x^2 - x - 8 = 4$, which simplifies to $x^2 - x - 12 = 0$. Factoring gives $(x - 4)(x + 3) = 0$, so $x = 4$ and $x = -3$.

Property

Problem-Solving Strategy

1. Read the problem to understand it.
2. Identify what you are looking for.
3. Name the unknowns with variables.
4. Translate the words into an algebraic equation.
5. Solve the equation using appropriate techniques.
6. Check if the answer makes sense in the context of the problem.
7. Answer the question with a complete sentence.

Examples

• The product of two consecutive positive integers is 56. Find the integers. Let the integers be $n$ and $n+1$. The equation is $n(n+1) = 56$, or $n^2 + n - 56 = 0$. Factoring gives $(n+8)(n-7)=0$. Since the integers are positive, $n=7$. The integers are 7 and 8.

• A rectangular patio has an area of 80 square feet. The length is 2 feet more than the width. Find the dimensions. Let width be $w$. Length is $w+2$. $w(w+2) = 80$, so $w^2 + 2w - 80 = 0$. This factors to $(w+10)(w-8)=0$. The width is 8 feet and the length is 10 feet.