Synthetic Division Shortcut

Algebra 2 · Axiom Academy

A faster, more efficient method for dividing a polynomial by a linear factor x - c — using nothing but its coefficients. Let's divide a polynomial by a linear factor. Our example is 2x^3 - 5x^2 + 3x - 7 divided by x - 2 . The key insight: we only need to work with the coefficients . Pull them into a row, and put the divisor's number in a box on the left. The coefficients become the top row The divisor's root c goes in the box 2. Long Division vs Synthetic Division Synthetic division isn't a different algorithm — it's polynomial long division with the bookkeeping stripped away . Every power of x you write in long division is redundant once the columns are lined up. Strip the x 's and the sprawling long division collapses into a single compact tableau. the same four moves, on the coefficients alone 1. Bring down the first coefficient. 2. Multiply it by c and write the product in the next column. 3. Add the column. 4. Repeat steps 2–3 until the last column is added. Let's run the whole cycle on , with c = 2 . Bring down the leader, then march the multiply ↗ then add ↓ cycle across every column. Each sum in the bottom row becomes the next number to multiply. Bring down 2 . Then , and -5 + 4 = -1 . Next , and 3 + (-2) = 1 . Finally , and -7 + 2 = -5 . Bottom row: . The bottom row is the quotient's coefficients plus the remainder, one degree lower than the dividend: Quotient 2x^2 - x + 1 · remainder -5 .

This is the written version of the interactive lesson above. See the full Algebra 2 course.