When the polynomial x³ - 2x² + 5 is divided by x - 1, what is the remainder?

By the remainder theorem, the remainder when a polynomial $p(x)$ is divided by $x - a$ is $p(a)$. Here $a = 1$, so the remainder is $p(1) = 1^3 - 2(1)^2 + 5 = 1 - 2 + 5 = 4$. You can verify with synthetic division using coefficients 1, $-2$, 0, 5: bring down 1; then $-2 + 1 = -1$; then $0 + (-1) = -1$; then $5 + (-1) = 4$. The last number, 4, is the remainder. A common wrong answer is 2, which comes from evaluating at $x = -1$ instead of $x = 1$. Another is 5, from dropping the other terms. Evaluate at the root of the divisor, which is $+1$.