Question about roots in polynomials

Question

Is it possible for a polynomial of degree higher than or equal to 5 having no complex and real root at all?

Answer 1

No -- every polynomial of positive degree with coefficients in $\mathbb C$ has at least one root in $\mathbb C$.

This is the Fundamental Theorem of Algebra.

When the degree is $5$ or more, the roots cannot necessarily be expressed with $\sqrt[n]{\vphantom X\cdots}$ signs (in combination with other arithmetic operations, starting from the coefficients), but they'll be out there somewhere nevertheless.