Factoring algebraic polynomials that are neither cyclic nor symmetric, and don't have obvious zeros

Question

In a set of $40$ problems, I was not able to factor these three polynomials. (The polynomials are neither cyclic nor symmetric, and don't have obvious zeros.) Any help is appreciated:

1) $x^3+2 \sqrt{3} x^2-2 \sqrt2.$

2) $x^3+x^2(1+y)-x(20+y)+20y.$

3) $x^3+(5-a)x^2+(8-3a)x-2a+4.$

Answer 1

The first two are irreducible over the fields generated by their coefficients. The third does factor. Hint: for a linear factor $x+c$, $c$ must divide the constant term.