I was trying this problem from my Abstract Algebra book exercise that says:
Show that the polynomial $x^2+\frac 13x-\frac 25$ is irreducible in $\mathbf Q[x]$.
What I tried: $x^2+\frac 13x-\frac 25 \equiv 15x^2+5x-6=f(x)$,say.
Now I compute ,$f(x+1)=15x^2+35x+14$. Now, $ 7 \mid 35, 7 \mid 14; 7\nmid 15,7^2 \nmid 14$. Hence using Eisenstein's criterion ,$f(x+1)$ is irreducible over $\mathbf Q$. Hence $f(x)$ is irreducible over $\mathbf Q$.
Am I right?
I want to add another problem similar to the above :
Show that the polynomial $2x^3-x^2+4x-2$ is not irreducible in $\mathbf Z[x]$.
Here,$f(x+1)=2x^3+5x^2+8x+3$ where $f(x)=2x^3-x^2+4x-2$. Here ,I can not apply Eisenstein's criterion for $f(x+1)$. I can not also factorize $f(x)$.Then how Can I prove it?
Can someone point me in the right direction? Thanks in advance for your time.