When x^13 + 1 is divided by x - 1, then what is the remainder?

Concept: The Remainder Theorem states that when a polynomial P(x) is divided by a linear divisor (x − c), the remainder equals P(c) — the value the polynomial takes at x = c. This follows from writing P(x) = (x − c)·Q(x) + R: substituting x = c makes the first term vanish, leaving R = P(c).

1. Identify the divisor x − 1, so the substitution value is c = 1.

2. Write the polynomial being divided: P(x) = x13 + 1.

3. Substitute x = 1 into P(x): P(1) = 113 + 1.

4. Since 1 raised to any power is 1, this simplifies to 1 + 1 = 2, so the remainder is 2.

Cross-check: For any polynomial of the form xn + 1 divided by x − 1, the remainder is always 1 raised to the power n, plus 1 — that is always 1 + 1, since 1 raised to any power stays 1, whatever n is. Had the divisor instead been x + 1, the substitution value would be x = −1, giving (−1)13 + 1 = −1 + 1 = 0 — a different result, which confirms the sign of the divisor's constant term matters and here the divisor is x − 1, not x + 1.