If $A=\begin{bmatrix}2&1&1\\0&1&0\\1&1&2\end{bmatrix}$, find $A^8-5A^7+7A^6-3A^5+A^4-5A^3+8A^2-2A+I$ 
If $A=\begin{bmatrix}2&1&1\\0&1&0\\1&1&2\end{bmatrix}$, find $A^8-5A^7+7A^6-3A^5+A^4-5A^3+8A^2-2A+I$

Attempt $1:$ $A=I+B$, where $B=\begin{bmatrix}1&1&1\\0&0&0\\1&1&1\end{bmatrix}$
$B^n$ comes out to be $2^{n-1}B$
After that I tried doing $A^8=(I+B)^8$ and then opening RHS with binomial. But it didn't help.
Attempt $2:$ Finding characteristic equation and putting $A$ in it. It didn't help either.
The equation I got here is $A^3-5A^2+7A-3I=0$
Note: Concept of minimal polynomial is not in syllabus. Also, if we can do this without characteristic polynomial, that would be great.

Hint:
$$A^5(A^3-5A^2+7A-3I)+A(A^3-5A^2+7A-3I)+A^2+A+I=\cdots$$
