First, a complaint about terminology. Many people write just as you do- that a "root" of a polynomial makes that polynomial 0. Strictly speaking a "zero" of a polynomial is a value of the variable that makes the polynomial equal to 0. A "root" of an equation is a value of the variable that satisfies the equation. But, so many people confuse the two, I'm probably fighting a losing battle!
A Polynomial is "irreducible" over over a field (typically the rational numbers or some extension of the rational numbers) (of course, it must have coefficients in that field) if it cannot be factored into factors with coefficients in the field. Since if a is a zero of a polynomial, then the polynomial has x- a as a factor, that is exactly the same as saying that a polynomial is irreducible if it has no zeros in the field.
For example is irreducible over the rational numbers because it cannot be factored as (x-a)(x-b) for any rational a and b.
It is, however, reducible over , the smallest field containing all rational numbers and because that field contains which is a zero. And that means it can be factored as .