SOLUTION: Let P and Q be polynomials such that P(x) and Q(P(Q(x))) have the same roots. If the degree of P is 7, what is the degree of Q?

Question 252551: Let P and Q be polynomials such that P(x) and Q(P(Q(x))) have the same roots. If
the degree of P is 7, what is the degree of Q?
Answer by drk(1908) (Show Source): You can put this solution on YOUR website!
Let P(x) = X^7 and Q(x) = X.
Right now the real roots of P(x) = 0.
P(Q(x)) = X^7
Q(P(Q((x))) = x^7.
The roots of Q(P(Q(x))) = 0.
So it appears that the degree of Q = 1.
In general, if the roots are both 0, then the degree of Q >=1, hypothesis.
We could try to generalize this but that may take more time than what is necessary.
Hope this helps.

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