While practicing a couple of problems in equality, I realized that it would be rather handy to have a method which can help one to prove whether a polynomial of degree n with real coefficients is positive or negative. In the case of quadratics, it is quite trivial as one may factorize the expression or rewrite it after perfect squaring. So, for instance consider the following quaritic equation:-
Is there a way to prove that this function is always positive and in general is there a technique which could be used for all polynomials.
(Methods that involve Calculus will be really appreciated.)
PS:I think that finding the Range of the polynomial might to do the trick, if the lowest point(if it exists) is positive than the polynomial will always be positive.