A quadratic polynomial has sum of zeroes $\frac{1}{2}$ and product of zeroes −6.
Which polynomial with INTEGER coefficients has these zeroes?

Start from $x^2-(\alpha +\beta )x+\alpha \beta =x^2-\frac{1}{2}x-6$. Multiplying EVERY term by 2 clears the fraction without changing the zeroes: $2x^2-x-12$. (Check: sum $=-\frac{-1}{2}=\frac{1}{2}$, product $=\frac{-12}{2}=-6.)2x^2+x-12$ has sum $-\frac{1}{2}$; $2x^2-x+12$ has product +6; $2x^2-12x+1$ swaps the roles of the sum and the product.