What makes a polynomial irreducible?
Basically, a non-constant polynomial is irreducible if it cannot be factorised into non-constant polynomials.
(An example of a non-constant polynomial is $\displaystyle P(x)=x^2 + 3x+2$. An example of a constant polynomial (sometimes called trivial) is $\displaystyle P(x) = 3$.)
I'm not seeing anyone touch an the real (pun intended) answer.
Polynomials are never irreducible. Over the complex numbers.
They are only irreducible over the reals (or rationals, integers etc.).
A polynomial that is irreducible over the reals is one that has strictly complex roots. (note this will be an even degree polynomial. why? )
what are the roots of $x^4 + 64$ ?
w/o going into detail these are $x=\pm 2 \pm 2 i$
i.e. all the roots are complex
But $x^4 + 64$ factors into $(x^2 - 4x + 8)(x^2 + 4x + 8).$ That means it is
reducible over the reals. In particular, it is reducible over the integers.
** From above in the lower quote box. This polynomial has strictly complex
roots, but it is not irreducible.