Try $q(x) = (x^2+1)^3$ with $n = 6$. The following $5$ linearly independent polynomials $p(x)$ qualify: $$ 4 x - 3, x^2, 4 x^3 - 1, x^4 + 4 x^3, x^6 - 24 x^3 $$
Robert Israel
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