Okay, here we have this:
Considering the provided polynomials, we are going to calculate the requested value, so we obtain the following:
So first we will perform the division assuming that k=0 to see what is the remainder that is obtained:
[tex]\begin{gathered} \frac{x^2-x}{x-1} \\ =\frac{x(x-1)}{x-1} \\ =x \end{gathered}[/tex]We obtain that the remainder when taking k=0, is zero, then it will mean that the k must be equal to the residue that we want:
In other words, if we want the remainder to be 3, k must be equal to 3, replacing:
[tex]\begin{gathered} \frac{x^2-x+k}{x-1} \\ \frac{x^2-x+3}{x-1} \\ =x+\frac{3}{x-1} \end{gathered}[/tex]Finally we confirm that for the remainder to be 3, the value of k must be equal to 3.