Find coefficients $a,b,c$ and the roots of polynomial:
$$f(x)=ax^5+3x^4+bx^3+4x^2+3x+c$$
with integer coefficients, if it's known that $f$ has triple integer root.
(I'm not sure if that was the correct translation to english so my apologies for any mistakes).
What I tried: I found first and second derivation:
$$f'(x)=5a^4+12x^3+3bx^2+8x+3$$ $$f''(x)=20ax^3+36x^2+6bx+8$$
If polynomial $g(x)=a_nx^n+...+a_1x+a_0$ has integer root $k$, then $a_0$ is divisible by $k$. That means that the triple integer root of $f$ divides both of the constant coefficients of both $f'(x)$ and $f''(x)$, $3$ and $8$.
That means that the only possible solution for $k$ is $k=\pm1$.
Do I now do two cases, each with different $k$, and solve the three equations $f(x)=0$, $f'(x)=0$ and $f''(x)=0$ and see which one of them is correct or is there another way to immediately know which of the two possible roots is the correct one? Thanks in advance!