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Yesterday, I wrote a post on the Mason-Stothers theorem and presented an elementary proof of the theorem given by Noah Snyder. As mentioned in that post, I will present now a problem proposed by Magkos Athanasios (*Kozani, Greece*) that can be solved almost “effortlessly” using the aforesaid theorem.

Problem: Let and be polynomials with complex coefficients and let be a complex number. Prove that if

for all , then the polynomials and are constants.

(*Magkos Athanasios*)

**Solution**: First, note that if is a constant, then this forces to be a constant, and vice-versa. Now, suppose and are *not* constants. We show that this leads to a contradiction.

Observe that if and have a common root, say, , then we have , which implies , which implies , a contradiction. Therefore, we conclude and are relatively prime polynomials, and hence, and are also relatively prime. Now, let and . Then, from the given equation, we conclude and for some .

So,

.

Also,

.

Now, applying the Mason-Stothers theorem, we get

, which implies , a contradiction! And, we are done.

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