Polynomial of degree 5 such that $P(x)-1$ is divisible by $(x-1)^3$ and $P(x)$ is divisible by $x^3$ 
$P(x)$ is a polynomial of degree 5 such that $P(x)-1$ is divisible by $(x-1)^3$ and $P(x)$ is divisible by $x^3$. Find $P(x)$.

No idea where to start, would $P(x)$ be of the form $x^3(Ax^2+Bx+C)$?

$P(x)=1+(ax^2+bx+c)(x-1)^3=1-(c+bx+ax^2)(1-3x+3x^2-x^3)$
$=1-c+x(3c-b)+x^2(-a+3b-3c)+x^3(\cdots)$
We need $1-c=3c-b=-a+3b-3c=0$   
