1 - x + 6x3
-7
x
are all examples of polynomials. Here's a precise definition.
A polynomial in an indeterminate (or variable x) with
coefficients in a ring R is a formal expression
f=a0 + a1x + a2x2 + ... + anxn
where
n is a nonegative
integer, and each
ai is in
R.
The
constant term of
f is
a0 and
ai
is called the
coefficient of
xi in
f.
The coefficients
of
xN in
f, for
N>n are defined to be zero.
Given a second formal expression
g=b0 + b1x + b2x2 + ... + bmxn
Then we say that
f=g if and only if for each
i the coefficents of
xi in
f and
g are equal. (This is what we mean when we speak
of
equating coefficients in a polynomial equation.)
The polynomial f is called zero if all its coefficients are
zero. A nonzero polynomial has a degree which is defined to
be the largest nonnegative integer t such that the coefficient
of xt in f is nonzero. By convention
the zero polynomial has degree -infinity. For short we write
deg f for the degree of f.
We define rules to add and multiply the polynomials f,g as follows
f + g = (a0+b0) + (a1+b1)x + ... + (an+bn)xn + ... +bmxm
if
m>=n and
f . g = c0 + c1x + c2x2 + ... + cn+mxn+m
where
ck=a0bk + a1bk-1 +...+ arbk-r +...+ a0bk
These rules makes the collection of all such polynomials
into a ring. This ring is denoted by R[x] and is
called the polynomial ring in x with coefficients
in R. Note that the collection of constant terms is
a subring of R[x] which is isomorphic to R.
We always identify R with this subring and think of R as sitting
inside R[x].
Note that deg(f+g) <= max{ deg f, deg g }.
If R is an integral domain then deg(fg) = deg f + deg g.
Polynomials in several variables
Starting with R[x] we could form the polynomial ring
R[x][y]
in a new indeterminate, y, say.
The elements of this ring have a unique expression
as a sums of terms
ai,jxiyj
with
(i,j) ranging over finitely many pairs of nonnegative inetgers and
ai,j nonzero in
R.
Clearly, if we had started with R[y] and then formed
R[y][x] we would have obtained the same ring.
This ring is denoted by R[x,y]
More generally, we can inductively define
R[x1,...,xn]=R[x1,...,xn-1][xn]
the polynomial ring in several indeterminates. A typical element
of this ring can be written uniquely as a sum of
monomials
ai1,i2,...,inx1i1...xnin
where
(i1,...,in) range over finitely many
tuples of integers and
ai1,i2,...,in
is nonzero in
R.
The total degree of such a monomial is defined to be
i1+...+in and the total degree
of a nonzero polynomial is the largest total degree of a monomial.