<s>[INST] Polynomial Division - "Define the largest natural number..." Would someone mind helping me with this question? The more detailed possible so I can have 100% of understanding. Thanks.  
Question: Define the largest natural number m such that the polynomial
$$P(x) = x^5-3x^4+5x^3-7x^2+6x-2$$ 
be divisible by $(x-1)^m$.
 [/INST] Using the Euclidean division of $x^5-3x^4+5x^3-7x^2+6x-2$ and $x-1$ we get:
$$x^5-3x^4+5x^3-7x^2+6x-2=(x^4-2x^3+3x^2-4x+2)(x-1)$$
Then apply the Euclidean division of $x^4-2x^3+3x^2-4x+2$ and $x-1$.
Then we get $x^4-2x^3+3x^2-4x+2=q(x-1)$.
Then apply the Euclidean division of $q$ and $x-1$ and so on.
</s>