[Math] Roots of a polynomial with irrational coefficients

Question

I know about algebraic numbers and transcendental numbers.
How the roots of a polynomial with irrational coefficients are classified.
Are they transcendental?

Answer 1

The roots of a polynomial with algebraic coefficients are all algebraic, and a monic polynomial whose roots are all algebraic has algebraic coefficients.

So a monic polynomial with some transcendental coefficient must have at least one transcendental root (and vice versa), but it can also have algebraic roots (for example, $0$ is a non transcendental root of $X^2- \pi X = 0$).