Factor x^4 + 1 with x-1 step by step. This solution is basically the same as that of the long division.
x^4 +1 = x^3(x - 1) + x^3 +1
= x^3(x - 1) + x^2(x - 1) + x^2 + 1
= x^3(x - 1) + x^2(x - 1) + x(x - 1) + x + 1
= x^3(x - 1) + x^2(x - 1) + x(x - 1) + 1·(x -1) + 1 + 1
= (x - 1)(x^3 + x^2 + x + 1) + 2
CK: (x - 1)(x^3 + x^2 + x + 1) + 2 = x(x^3 + x^2 + x +1) - (x^3 + x^2 + x+ 1) + 2 = x^4 + 1
Substitute x=1 in x^4 + 1. We have: (1)^4 + 1 = 2 Therefore, the remainder when x^4 + 1 is divided by x - 1 is 2. (the remainder theorem) CKD.
The answer: (x^4 +1)/(x-1) = x^3 +x^2 + x +1 r. 2