EQUATIONS AND VARIETIES 61

Finding Roots of Polynomials

The easiest general class of varieties to look at would be those

deﬁned by a single Z-equation in a single variable, for instance,

x

3

+ x − 2 = 0. (6.2)

The study of this type of variety is dominated by the concept of the

Galois group (see chapters 8 and 13).

We now review the following terminology:

DEFINITION: If f (x) is a polynomial, the roots of f (x)are

those numbers c so that f (c) = 0.

The roots of f (x) are the solutions to the polynomial equation

f (x) = 0. Now, there is the particularly simple equation x

n

= a,and

we call a solution to it an nth root of a. Sometimes “root” means a

square root or cube root, and so on, and sometimes it means a root

of a more general polynomial. You should be able to tell which root

is meant from the context.

It is helpful to symbolize the polynomial we are studying by

a single letter, say p. If we want to remember the name of the

variable, we can write p(x). For instance, p(x) might denote the

polynomial x

3

+ x − 2. Then equation (6.2) can be written p(x) = 0.

This looks like functional notation, and it is. If x is a variable, p(x)

just stands for the polynomial p,butifa is a number, p(a)stands

for the number you get by substituting a for x in p. For instance, if

p(x) = x

3

+ x − 2thenp(0) = 0

3

+ 0 − 2 =−2, p(1) = 1

3

+ 1 − 2 = 0,

and in general, p(a) = a

3

+ a − 2.

For example, consider the variety of solutions, call it S,tothe

equation p(x) = 0, where p(x) is the polynomial discussed in the pre-

ceding paragraph. That is, if A is any number system, describe the

set S(A), which is the set of all elements a of A such that p(a) = 0.

We can use simple algebra and some guessing to ﬁnd S(Z): We

have just seen that p(1) = 0. So S(A) contains the number 1. High-

school algebra now tells us that we can divide p(x)byx − 1andwe

are guaranteed that it will go in without remainder. Doing that we

get the quotient x

2

+ x + 2.

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