- Topics At a Glance
- Exponents
- Negative Exponents
- Fractional Exponents
- Irrational Exponents
- Variables as Exponents
- Defining Polynomials
- Degrees of a Polynomial
- Multivariable Polynomials
- Degrees of Multivariable Polynomials
- Special Kinds of Polynomials
- Evaluating Polynomials
- Roots of a Polynomial
- Combining Polynomials
- Multiplying Polynomials
- Multiplication of a Monomial and a Polynomial
- Multiplication of Two Binomials
- Special Cases of Binomial Multiplication
- General Multiplication of Polynomials
- We'll Divide Polynomials Later!
- Factoring Polynomials
- The Greatest Common Factor
- Recognizing Products
- Trial and Error
- Factoring by Grouping
- Summary
**Introduction to Polynomial Equations****Solving Polynomial Equations**- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

To solve a polynomial equation, we need to find the values of *x* that make the polynomial 0. That is, we want to find the roots of the polynomial. Then, if we want to give it a root canal, we'll know where to start.

If the polynomial factors into polynomials of degree 1, we can find the roots by factoring the polynomial. Ah, it feels good to stretch our factoring muscles once again. It's been five minutes; they were starting to cramp.

Solve *x*^{2} – 4*x* – 5 = 0.

We can factor the polynomial as

(*x* – 5)(*x* + 1).

In order for this product to equal zero, one of the factors must equal zero. Either

*x* – 5 = 0 and so *x* = 5

or

*x* + 1 = 0 and so *x* = -1.

The roots of the polynomial, which are the solutions to the equation, are *x* = 5 and *x* = -1.

When a polynomial doesn't factor nicely, it can be hard to find its roots, even if you do extensive research on Ancestry.com. We'll talk about ways to find roots for some other polynomials later, so hold onto your hat.

Exercise 1

Solve *x*^{3} - 5*x*^{2} + 6*x* = 0.

Exercise 2

Solve *x*^{2} - 25 = 0.

Exercise 3

Solve - *x*^{2} - 8*x* + 9.