$f(x) \mid g(x) \Rightarrow f(x^k) \mid g(x^k)$

Question

How to prove :

$$f(x) \mid g(x) \Rightarrow f(x^k) \mid g(x^k)$$

Note: Just tagging this as being about polynomials requires a lot of extra effort to figure out you are talking about polynomials. — Thomas Andrews, Nov 30 '13 at 16:05

Calvin Lin · Answer 1 · 2013-11-30T16:09:19.120

6

Hint: By the definition, $f(x) \mid g(x)$ if and only if there is a polynomial $h(x) $ such that $g(x) = f(x) \times h(x)$.

So, the initial condition in the question tells us that we have a $h(x)$ such that $g(x) = f(x) \times h(x)$.

Can you think of a polynomial $H_k(x)$ such that $g(x^k) = f(x^k) \times H_k(x)$? How does $H_k(x)$ relate to $h(x)$?

edited Nov 30 '13 at 16:09

answered Nov 30 '13 at 15:56

Calvin Lin

1 Answers1