For polynomials with integer coefficients the question is roughly the same as "what are the practical applications of algebraic number theory". The usual answers are coding theory and cryptography where factorization (and related operations such as testing whether a polynomial can be factorized) is part of the basic infrastructure from which systems are built or broken. Coding is necessary for digital communication (including telephone, video and satellites) and cryptography has become a basic feature of everyday computer use and commerce.
For polynomials with real coefficients there is partial fraction expansion used in calculus to compute integrals.
For polynomials with complex numbers as coefficients the factorization is into linear factors so that factoring is practically the same as numerical root finding (and this is in part true for real numbers as well). Problems in engineering where the location of complex roots of a polynomial determines the behavior of the system are common. For example, stability or instability can be decided by whether all the roots are inside the unit circle, or have positive real part, or other location-based criteria. Oscillations might be periodic if roots are n'th roots of 1 for some n, or quasiperiodic behavior if roots are on the unit circle but not all at roots of 1. A system governed by a partial differential equation would show diffusion (like heat) or wave-like behavior based on the factorization of an associated "differential operator", which is essentially a polynomial.
In general, many phenomena are decomposable into components, pieces or subsystems in a way that (when the systems are modeled mathematically) appears as a multiplicative decomposition of I don't have any idea what this fraction 1848 / 16632 means.
The numbers are too large. But if I factor the numerator and
denominator I get: (3)(8)(7)(11) / (8)(11)(7)(9) and I can
see that the 3, the 8, the 7, and a 5 are factors of both the
numerator and denominator. Since 3/3, 8/8, 7/7 and 11/11 are all
equal to 1 the fraction reduces to 1/3, which is a lot easier to
understand and to compute with.
It's the same with complex algebraic fractions. (x^4 - x^2)/(x^3+x^2)
looks REALLY complicated, but the numerator factors into x^2(x-1)(x+1)
and the denominator factors into x^2(x+1), and since x^2/x^2 and
(x+1)/(x+1) are both equal to 1, this fraction simplifies to x-1 . . .
a LOT easier to work with.
A second important use of factoring is in solving equations. You don't
need to factor to solve 2x+3 = 5 ... linear equations use a different
method. And you don't need to factor second degree equations because
you can use the Quadratic Formula (although factoring is often MUCH
easier!). But if you need to solve equations where the degree of the
highest term is more than 2 then you really have no choice at all
because you don't have formulas for most of them.
Here is an example of a hard equation to solve:
x^5 - 4x^4 - 12x^3 = 0
But if you factor it completely you get (x^3)(x-6)(x+2) = 0 and now it
is easy, because this says that a product of things turns out to be
equal to zero. If you multiply, the only way to get zero as an answer
would be if you multiplied by zero. So one of the three factors has
to be zero.
If x^3 = 0 then x = 0
If x-6 = 0 then x = 6
If x+2 = 0 then x = -2
So the solutions to this equation are x = 0 or 6 or -2.
