Study Guide

Polynomials - Introduction to Polynomial Equations

Introduction to Polynomial Equations


There are two different definitions of a polynomial equation that show up in books, on websites, and in bathroom stalls, but the two definitions actually mean the same thing. Here, we'll prove it. In the future though, please stop reading bathroom stalls. We'd like you to hold onto what's left of your innocence.

Definition 1: A polynomial equation is any equation that can be written in the form (polynomial) = 0.

Here's an example of the first type: x7 + 8x – 43 = 0. Pretty basic stuff, right?

Definition 2: A polynomial equation is any equation that sets one polynomial equal to another:

(one polynomial) = (other polynomial)

An example of the second type would be something like x2 + 3x = 8x + 17.

Any equation of the form (polynomial) = 0 is basically the same as setting one polynomial equal to another, since 0 is a polynomial. Anything that counts as a polynomial equation according to the first definition is also a polynomial equation according to the second definition. In other words, the first definition is definitely defined by the definitive definition of the...huh. We should have stuck with the first way we said it.

On the other hand, any equation that sets one polynomial equal to another can be written in the form (polynomial) = 0.

For example, the polynomial equation 7x = 5x can be written as 2x = 0 after we subtract 5x from both sides.

Any polynomial equation under the second definition is also a polynomial equation under the first definition.

The short story is that you can pick whichever definition makes more sense to you and go with that. For now, we'll use Definition 1, which is more convenient for the next section. However, just because we're using it for the time being doesn't mean it's fundamentally any better; we love all our definitions equally.

  • Solving Polynomial Equations


    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.

    Sample Problem

    Solve x2 – 4x – 5 = 0.

    First we factor the polynomial:

    (x – 5)(x + 1)

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

    x – 5 = 0
    x = 5

    Or:

    x + 1 = 0
    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.

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