Degree reduction

In $$\mathbb Z_p$$ find a polynomial $$r$$ of degree at most $$p$$, that attains the same values as the polynomial $$q$$ for all $$x\in \mathbb Z_p$$.
In $$\mathbb Z_5$$, $$q(x)=4x^{20}+3x^{17}+2x^{16}+x^{13}+3x^{12}+2x^{10}+4x^9+2x^7+2x^5+x+3$$.
In $$\mathbb Z_7$$, $$q(x)=5x^{16}+6x^{15}+4x^{13}+x^{12}+3x^{11}+6x^{10}+2x^9+3x^7+5x^5+2x+1$$.