3
Factors and Zeros
3.1 Irreducible Polynomials
30 = 6.5; t3-6t+4 = (t-2)(t2+2t-2). Both equationsexpress an element
as a product of others. If we disallow the use of +l and -1, the factors of
30 are smaller than 30, so that 30 can be written as a nontrivial product
of integers in only finitely many ways. Furthermore, factoring further gives
30 = 2.3 .5 and every factorization of 30 involves products of the primes
2, 3, 5 or their negatives.
For polynomials, degree plays the role of numerical size in restricting
the ways in which a polynomial can be written as a product of others.
This is a similarity between the domains of integers and polynomials which
distinguishes each from the fields of rationals, reals and complex numbers.
Specifically, we ask:(a) To what extent can a polynomial be decomposed as a product of other
polynomials? Is it ever possible to continue factoring the factors we
get indefinitely, or must we stop after a finite amount of time?(b) Is there a notion of “prime” analogous to that for number which can
be applied to polynomials?(c) Can every polynomial be written as a product of these “prime” ones?
If so, is such a representation unique up to order of factors?(d) Can we actually identify the “prime” polynomials?Let us look in turn at these equations. Over a field, such as Q, R and
C, if p(t) is a polynomial and c is a nonzero constant, then c-lp(t) is
also a polynomial over the field. Thus, every polynomial admits trivial
factorizations of the type
p(t) = c. c-‘p(t).The constant polynomials play the role of +l and -1 for the integers in
that they are universal divisors. If we are to give a meaningful analysis of
the questions asked, we should ask them only in the context of nontrivial
factorizations.
If p = fg is a nontrivial factorization of p, then deg f and degg are both
strictly less than deg p, and deg p = deg f + deg g. We cannot continue to