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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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