O'Reilly logo

Fearless Symmetry by Robert Gross, Avner Ash

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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).

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required