Find the remainder when x³ − 2 is divided by x − 1.

Step-by-step solution

1. 1

   Use the Remainder Theorem rather than long division. If a polynomial $f(x)$ is divided by $x - a$, the remainder is simply $f(a)$. Here the divisor is $x - 1$, so $a = 1$.

2. 2

   Evaluate $f$ at that value.

   $f(1) = 1^3 - 2 = 1 - 2 = -1$

3. 3

   Sanity-check the form of the answer. The divisor $x-1$ has degree $1$, so the remainder must have degree $0$ — a plain constant. Any answer containing $x$ or $x^2$ can be ruled out immediately, which is a fast way to eliminate distractors on a multiple-choice question.