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