88 CHAPTER 8

Polynomials and Their Roots

We review here the basic facts and deﬁnitions about polynomials.

1. A Z-polynomial is a polynomial in one variable with

integer coefﬁcients.

2. The degree of a Z-polynomial is the degree of the highest

power of the variable that occurs. For example, the degree

of 11x

5

+ 103x − 41 is 5.

3. Any Z-polynomial f with degree at least 1 can be factored

into a nonzero constant c times the product of factors of the

form (x − a), where a is some complex number.

4. If the degree of f is d, then there are exactly d factors.

5. The roots of f are deﬁned to be those complex numbers b

such that f (b) = 0. We can think of the equation f (x) = 0as

deﬁning a variety, S

f

, and then the roots of the polynomial

are the elements of S

f

(C).

The roots of f are exactly the a’s appearing in its factorization

into terms of the form (x − a). This is easy to prove. Suppose that

we have factored

f (x) = c(x − a

1

)(x − a

2

) ···(x − a

d

).

The number of factors on the right-hand side, d, must be the

same as the degree of f (x). If we replace x by any of the numbers

a

1

, a

2

, ..., a

d

, throughout the equation, the right-hand side multi-

plies out to be 0, so the left-hand side must also be 0; that is,

f (a

1

) = f(a

2

) =···=f (a

d

) = 0. On the other hand, if we substitute

in some number t which is different from all the a

k

’s, then the right-

hand side is a product of nonzero numbers, and so must be nonzero.

This says that f (t) = 0. Thus, we have shown that f (t) = 0ifand

only if t is one of the numbers a

k

.

Before we give some examples, notice that a factor of the form

(x + b) is really of the form (x − a) where a =−b. For instance,

(x + 3) = (x − (−3)). So when we factor f (x) into linear factors, we

will feel free to use factors of either form: (x − a)or(x + b).

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