α and β are the zeroes of $x^2+px+q$, where $q$ ≠ 0. Which polynomial has 1/α and 1/β as its zeroes?

$\alpha +\beta =-p$ and $\alpha \beta =q$. New sum $=\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }=-\frac{p}{q}$; new product $=\frac{1}{\alpha \beta }=\frac{1}{q}$. So the polynomial is $x^2+\frac{p}{q}x+\frac{1}{q}$, and multiplying by $q$ gives $q{x}^2+px+1$ — the original coefficients simply read backwards. $q{x}^2-px+1$ flips the sign of the new sum; $p{x}^2+qx+1$ reverses only the first two coefficients; $x^2+qx+p$ swaps $p$ and $q$ without reversing the whole list.