Let $p(x)$ be a polynomial of degree $3$ with real coefficient.
Then which of the following is possible $?$
$A. p(x)$ has no real root. WRONG.
$B.p(x)$ has exactly two real roots. WRONG AGAIN.
$C.p(1)=-1,p(2)=1,p(3)=11\ and\ p(4)=35.$
$D.i-1\ \ and\ \ i+1$ are roots of $p(x)$. As these are NOT conjugates, so cannot be roots of the same polynomial so AGAIN WRONG .
So, that leaves out only option $C$. Now , what I was thinking was that , I could say only $C$ is possible because I had ways of eliminating the rest . But what if the eliminations were not this easy , was there any way to say that , a polynomial of degree $3$ actually exists satisfying what is given in $C$ $?$ How can I actually find out such a polynomial $?$
Hope I could convey my question properly.
Thanks for any help.