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Study Guide

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Evaluating a polynomial is like evaluating any other expression. We substitute the given value(s) for each and every variable. Then we do the necessary plusing and minusing to find an answer. If you come across a polynomial that involves only subtraction, you may be a little nonplussed.

Evaluate the polynomial 4*x*^{2} – 2*x* + 7 for *x* = 3.

We substitute 3 for every occurrence of *x* to get 4(3)^{2} – 2(3) + 7 and simplify to 37.

That really wasn't as bad as you were expecting, was it?

### Roots of a Polynomial

The**roots**of a polynomial are the values of*x*(or whatever variable shows up in the polynomial) that make the entire polynomial have a value of zero when we evaluate the polynomial at those values. Think of it this way: there are two "0"s in the word "root." Fine, they're actually "o"s, but we won't tell anybody if you don't.### Sample Problem

What are the roots of the polynomial

*x*^{2}– 9?We're looking for the

*x*-values that turn the whole polynomial into a big, fat zero, so let's set it equal to zero and solve for*x*.*x*^{2}– 9 = 0*x*^{2}= 9*x*= ±3In other words, we can plug in either

*x*= 3 or*x*= -3 to make our polynomial have a value of zero. This means 3 and -3 are the roots of the polynomial*x*^{2}– 9.

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