Question 157
Marcella, while playing, finds purely by chance two polynomials $p(x)$ and $q(x)$, non-constant and with integer coefficients, satisfying the relation:
$$
p(q(x+1)) = p\left(x^{3}\right) q(x+1)^{5}.
$$
What can we state with certainty about the two polynomials found by Marcella?
(A) The leading coefficient of $p(x) q(x)$ is positive
(B) The polynomial $q(x)$ has no integer roots
(C) The degree of $q(x)$ does not exceed that of $p(x)$
(D) The sum of the coefficients of $p(x)$ is odd
(E) The degree of $p(x) q(x)$ is a multiple of $8$
$$
p(q(x+1)) = p\left(x^{3}\right) q(x+1)^{5}.
$$
What can we state with certainty about the two polynomials found by Marcella?
(A) The leading coefficient of $p(x) q(x)$ is positive
(B) The polynomial $q(x)$ has no integer roots
(C) The degree of $q(x)$ does not exceed that of $p(x)$
(D) The sum of the coefficients of $p(x)$ is odd
(E) The degree of $p(x) q(x)$ is a multiple of $8$