Factorising $D^{2n} - I$ and $D^n - I$ $\rm(ii)$ Let $\mathcal P_{11}(\Bbb R)$ be the vector space of polynomials of degree $\leqslant11$ over the field $\Bbb R$ and let $D:\mathcal P_{11}(\Bbb R)\to\mathcal P_{11}(\Bbb R)$ be the linear map given by differentiation. Write down the least positive integer $n$ for which $D^{2n}=0$ on $\mathcal P_{11}(\Bbb R)$.  By factorisation of the formal expressions $D^{2n}-I$, $D^n-1$ or otherwise, show that the mapping $$D^9-D^6+D^3-I:\mathcal P_{11}(\Bbb R)\to\mathcal P_{11}(\Bbb R)$$
I can't seem to get a factor of $D^9-D^6 + D^3 - I$,could someone show me how to get the factor they require? I'm assuming it's in the form of $D^{12}-I$, but all I have is $D^{12} - I = (D^4-I)(D^8+D^4+I)$ by guess work

Hint: Use polynomial division. 
