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**A6363** Is $\displaystyle a=3x^3+2x+2$ reducible in the rationals?

So $\displaystyle a$ is reducible iff we can write $\displaystyle a=bc$, for $\displaystyle bc$ in the rationals (polynomials) and b,c$\displaystyle \neq$0. So deg(a)=3, therefore 0<deg(b)<deg(a) and 0<deg(c)<deg(a). So either deg(a)=2 or deg(a)=1, since deg(b)+deg(c)=deg(a)=3

The next step I'm not too sure about, but I believe it is to show that a(x)$\displaystyle \neq$0 for any x in the rationals. And this leads to the conclusion that a is irreducible since we cannot write a=bc since either b or c is a unit.

Any help will be appreciated, sorry that my notation is not entirely accurate in this post (not used to LateX).