Descartes' rule of signs claims that there exist an upper bound on the number of real roots of a polynomial based on the sign differences between consecutive nonzero coefficients.
But, there exists some similar criterion to obtain a lower bound on the number of real roots?
P.S. It's clear that for polynomials with odd degree, a real root is guaranteed.
As @Andy Walls mentioned in his comment, an even-degree polynomial with real coefficients is not guaranteed any real roots - $y=x^2+1$ is a simple example. If the polynomial is of odd degree and has real coefficients, you can prove by the Intermediate Value Theorem that it has at least one real root, call it $a$. But then, if you factor out the expression $x-a$ from the polynomial, what's left is of even degree, and itself is not guaranteed any real roots.
If you have a polynomial of even degree with real coefficients, and an odd number of sign changes in those coefficients, then you're guaranteed two real roots, because complex roots always come in complex conjugate pairs.
Summary: odds have minimum one root, evens have minimum zero roots unless there are an odd number of sign changes, in which case you're guaranteed two real roots.
All bets are off if you have complex coefficients.