Final answer:
The integers n for which the expression (n^3 + 1)/(n - 1) is an integer are n = 0, n = -1, and n = 2. These values ensure that the numerator is divisible by the denominator, and that division by zero does not occur.
Step-by-step explanation:
To find all integers n for which the expression (n^2 * n + 1)/(n - 1) is an integer, we must ensure the numerator is divisible by the denominator without any remainder. However, there seems to be a typo in the original expression and it might be (n^3 + 1)/(n - 1) instead.
When n = 1, the denominator becomes zero, and division by zero is undefined. Therefore, n = 1 is not a valid solution. When n = 0, the expression becomes (0 + 1)/(-1), which equals -1, an integer. Hence, n = 0 is a valid solution. When n = -1, the numerator becomes ((-1)^3 + 1) which equals 0, and since division of zero by any number (n - 1, in this case) is 0, n = -1 is also a valid solution.
Lastly, for n = 2, the expression becomes (2^3 + 1)/(2 - 1), which simplifies to 9/1, and that equals 9, an integer. Thus, n = 2 is a valid solution as well.
Based on these evaluations, the integers for which the expression is an integer are n = 0, n = -1, and n = 2.