# 6.5: Polynomial Equations

- Page ID
- 30868

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By the end of this section, you will be able to:

- Use the Zero Product Property
- Solve quadratic equations by factoring
- Solve equations with polynomial functions
- Solve applications modeled by polynomial equations

Are you ready?

Before you get started, take this readiness quiz.

We have spent considerable time learning how to factor polynomials. We will now look at polynomial equations and solve them using factoring, if possible.

A **polynomial equation** is an equation that contains a polynomial expression. The **degree of the polynomial equation** is the degree of the polynomial.

POLYNOMIAL EQUATION

A ** polynomial equation** is an equation that contains a polynomial expression.

The ** degree of the polynomial equation** is the degree of the polynomial.

We have already solved polynomial equations of **degree one**. Polynomial equations of degree one are linear equations are of the form ax+b=c.ax+b=c.

We are now going to solve polynomial equations of **degree two**. A polynomial equation of degree two is called a **quadratic equation**. Listed below are some examples of quadratic equations:

\[x^2+5x+6=0 \quad 3y^2+4y=10 \quad 64u^2−81=0 \quad n(n+1)=42 \nonumber\]

The last equation doesn’t appear to have the variable squared, but when we simplify the expression on the left we will get n2+n.n2+n.

The general form of a quadratic equation is \(ax^2+bx+c=0\), with \(a\neq 0\). (If \(a=0\), then \(0·x^2=0\) and we are left with no quadratic term.)

QUADRATIC EQUATION

An equation of the form \(ax^2+bx+c=0\) is called a quadratic equation.

\[a,b,\text{ and }c\text{ are real numbers and }a\neq 0\nonumber\]

To solve quadratic equations we need methods different from the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.