Synthetic Division

Synthetic division divides a polynomial by x − k using coefficients alone: bring down, multiply by k, add across, and read the quotient and remainder off the bottom row.

Divide x³ − 3x + 2 by x − 1, so k = 1. Coefficients with the missing term as zero: 1, 0, −3, 2. Bring down the 1. Multiply by k and add: 0 + 1 = 1. Again: −3 + 1 = −2. Again: 2 + (−2) = 0. Bottom row: 1, 1, −2, remainder 0 — the quotient x² + x − 2, matching the long division stroke for stroke at a fraction of the ink. The last number is always the remainder, and a zero there means x − k divides clean.