Unique Factorization for polynomials

[squelchy451]

Homework Statement

Prove unique factorization for hte set of polynomials in x with integer coefficients

Homework Equations

The Euclidean algorithm may be of some use

The Attempt at a Solution

Let's say that the polynomial is of the form anx^n + a(n-1)x^(n-1) ... a1x + a0

There are only n values of x not necessarily distinct so that the polynomial equals 0. If one of these values of x is p, then (x-p) divides the polynomial.

It's a more of a gut feeling/intuition thing that I need to prove using logic and actual theorems

 

[lippyka]

. To prove unique factorization, we need to show that if a polynomial is factorized into two or more factors, then those factors are unique. Let's assume that f(x) is the polynomial, and it can be factored as f(x)=g(x)h(x). We want to show that g(x) and h(x) are unique. We can use the Euclidean algorithm to find the greatest common divisor of g(x) and h(x), which we will denote as d(x). Now, if d(x)=1, then g(x) and h(x) are relatively prime and thus must be unique. If d(x) ≠ 1, then we can write g(x)=d(x)k(x) and h(x)=d(x)l(x). Now, if k(x) and l(x) are not relatively prime, then we can use the Euclidean algorithm again to find the greatest common divisor of k(x) and l(x). We can repeat this process until the greatest common divisor of the two polynomials is 1, which proves that g(x) and h(x) are unique.