# Polynomial Divisors

Let . Show that for any positive integers there exists unique numbers such that the polynomial is divisible by p(x)

- 13 August, 2012 -
- Article, Math Problems -
- Tags : algebra, analysis, number theory, titu
- 0 Comments

Let’s first consider the case, just as an example. We want to show that , or show as some multiple of .

Next, we consider the case

and finally, just as an example, the case

this suggests that such that

Informally, suppose that , then it must be the case that

This concludes our proof that , where .

As an extension, we will further show that . Let . Immediately, we see that holds. Informally, we show that . The body of the inductive steps is as follows

This concludes our proof that . Sequentially,