Polynomial Long Division

Polynomial long division runs the arithmetic algorithm on terms — divide leading by leading, multiply back, subtract, repeat — and ends in the Division Algorithm: dividend equals divisor times quotient plus remainder.

Divide x³ − 3x + 2 by x − 1. Write the dividend with its missing term visible: x³ + 0x² − 3x + 2. Leading over leading: x³/x = x²; multiply back and subtract, leaving x² − 3x. Repeat: x²/x = x, subtract, leaving −2x + 2. Once more: −2x/x = −2, subtract — remainder 0. Quotient: x² + x − 2. The Division Algorithm is the receipt: (x − 1)(x² + x − 2) multiplies back to x³ − 3x + 2 exactly, and a zero remainder announces a clean factor.