What is the remainder when $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-3{\mathrm{x}}^2+2\mathrm{x}+1$ is divided by x - 1? 

The remainder theorem is as follows:

Let p(x) be any polynomial of degree greater than or equal to one, and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

Thus the remainder is f(1). 

∴ $\mathrm{f}\left(1\right)=1^3-3\times 1^2+2\times 1+1$ 

= 1 - 3 + 2 + 1 

= 1