The degree of the polynomial $p(x) = x^2 - x^3 + x + 1$ is $\ldots \ldots \ldots . .$

Divide $x^{3}-6x^{2}+11x-6$ by $x^{2}-8x+27$.

Given that $x-\sqrt{5}$ is a factor of the cubic polynomial $x^{3}-3 \sqrt{5} x^{2}+13 x-3 \sqrt{5}$. Find all the zeroes of the polynomial.

If the polynomial $p(x) = x^{3} - 3x^{2} + x + 2$ is divided by the divisor polynomial $s(x)$, then the quotient polynomial $q(x) = x - 2$ and the remainder polynomial $r(x) = -2x + 4$ are obtained. Find the divisor polynomial $s(x)$.

Can $x-1$ be the remainder on division of a polynomial $p(x)$ by $2x+3$? Justify your answer.

In a prize-winning competition,$x+2$ students are qualified for the prize. If a total of $3x^3 + 10x^2 + 7x - 2$ prizes are distributed among them equally,find the number of prizes left undistributed,where $x \in N$.