High school pigeonhole principle question

Question

Let p(x) be a polynomial with real coefficients and such that the product of all the roots is negative. Show that if the degree of p(x) is 6, then p(x) has at least one positive root.

Answer 1

If a polynomial has real coefficients then for any complex root $z$, the conjugate $\bar{z}$, is also a root .

And $z\cdot \bar{z}=|z|^2>0$.

Therefore if the degree is $6$ it can have $0,2,4,6$ real roots, including multiple roots.

If the product of all roots is negative, then real roots can't be all negative: there must be a positive one, at least.