# Domain of irreducibility of $3x^3+20ax^2+50a^2x+60$

I need to evaluate for which $$a \in \mathbb{Z}$$ the polynomial $$3x^3+20ax^2+50a^2x+60$$ is irreducible respectively over $$\mathbb{Q}, \mathbb{C}$$ and $$\mathbb{R}$$. I think it is a tricky question but I need not to be wrong. For the fundamental theorem of algebra in $$\mathbb{C}$$ there should be $$n$$ roots, so it's always reducible. Every polynomial splits into at worst quadratic factors in $$\mathbb{R}$$.

Given the fact that

$$3x^3+a[20x^2+50ax]+60$$

if there was not the 60 I could find some $$a$$ such that $$3x^3 = -20ax^2$$ and get a polynomial of n=2 such that $$\Delta \lt 0$$ but it's not the case.

• If it's reducible (0ver the rationals), it has a rational root. If it has a rational root, the numerator of that root divides $60$, and the denominator divides $3$. So we've reduced it to a finite problem. Oct 5, 2020 at 12:06
• So, can you solve it now? Oct 7, 2020 at 10:25

I have finally got it (I think). Every cubic polynomial is reducible in both $$\mathbb{R}$$ and $$\mathbb{C}$$.
or you notice that you can apply Eisenstein's test with $$a_0 = 60$$, $$a_n = 3$$, and $$p=5$$ to find out it is irreducible.