The function $f(x)=x^{3}-x^{2}-x+1$ has at least one integer root. Use the integer root theorem to find that root, then proceed to find all complex roots. (Note: roots may be integer, rational, irrational, and/or complex.)
Answer Attempt 10 out of 100

Step 1 ：The integer root theorem states that if a polynomial has an integer root, then that root must divide the constant term of the polynomial. In this case, the constant term is 1, so the only possible integer roots are 1 and -1.

Step 2 ：We can test these roots by substituting them into the function: \(f(1) = 1^{3} - 1^{2} - 1 + 1 = 0\) and \(f(-1) = (-1)^{3} - (-1)^{2} - (-1) + 1 = 0\). So, both 1 and -1 are roots of the function.

Step 3 ：Now, we can use synthetic division to find the remaining roots. We'll start with 1. The remainder is not zero, so 1 is not a root. Let's try -1. The remainder is zero, so -1 is a root.

Step 4 ：The quotient is \(x^{2} - 2x + 1\), which is a quadratic that can be factored to \((x-1)^{2}\). So, the roots of the function are -1, 1, and 1.

Step 5 ：To check our work, we can substitute these roots back into the original function: \(f(-1) = (-1)^{3} - (-1)^{2} - (-1) + 1 = 0\) and \(f(1) = 1^{3} - 1^{2} - 1 + 1 = 0\).

Step 6 ：\(\boxed{\text{So, the roots of the function } f(x)=x^{3}-x^{2}-x+1 \text{ are -1, 1, and 1.}}\)