🧾 Synthetic & Polynomial Long Division
(x³ − 6x² + 11x − 6) ÷ (x − 2) = x² − 4x + 3, remainder 0 — so (x − 2) is a factor. The classic synthetic tableau, drawn for your numbers, plus full long division for any divisor.
Dividing x³ − 6x² + 11x − 6 by (x − 2):
| 2 | 1 | -6 | 11 | -6 |
| 2 | -8 | 6 | ||
| 1 | -4 | 3 | 0 |
x² − 4x + 3
quotient
0
remainder = p(2)
How to read the table
Bring the first coefficient down. Then repeatedly: multiply the bottom number by r (middle row) and add it to the next coefficient (bottom row). The last bottom number is the remainder — which by the remainder theorem equals p(r), and it's 0 here, so (x − 2) is a factor.
Exact rational arithmetic — r can be a fraction like 1/2 (testing the rational-root candidates). Runs locally.
How the synthetic & polynomial long division works
Synthetic division is the fast tableau for dividing by (x − r): bring down the leading coefficient, then repeatedly multiply by r and add to the next coefficient. The tool draws the three-row table exactly as it's taught, with the remainder highlighted — which by the remainder theorem equals p(r), so the same table doubles as polynomial evaluation and as a factor test (remainder 0 means (x − r) divides p). For divisors of higher degree, the long-division tab runs the full algorithm — divide leading terms, multiply, subtract, repeat — listing each step until the remainder's degree drops below the divisor's. Every coefficient is an exact rational, so r = 1/2 (a rational-root candidate) works precisely.
The remainder theorem readout is the underrated feature: to evaluate p(4), synthetic division by (x − 4) is faster and less error-prone than substituting powers — and the table is the working your algebra course wants shown. Testing rational-root candidates (±1, ±2, ±3, ±1/2…) one synthetic division at a time is exactly how cubic factoring is actually done by hand.
Frequently asked questions
How does synthetic division work?
Write r in the corner and the coefficients across the top. Bring down the first coefficient; multiply by r, write under the next coefficient, add; repeat. The bottom row is the quotient's coefficients, and the last number is the remainder.
What is the remainder theorem?
Dividing p(x) by (x − r) leaves remainder p(r). So synthetic division evaluates the polynomial as a by-product — and remainder 0 proves (x − r) is a factor, which the tool announces.
When do I need long division instead?
When the divisor isn't linear-monic — dividing by x² + 1 or 2x − 3 needs the full algorithm. The long-division tab handles any divisor degree with exact coefficients and the step list.
Can r be a fraction?
Yes — exact rational arithmetic means testing the rational-root candidate 1/2 gives precise coefficients (no 0.4999… drift), which matters because a near-zero float remainder is not a factor proof.
How do I enter missing terms?
Use 0 for absent powers: x³ − 1 is "1, 0, 0, -1". Coefficients are comma-separated, highest degree first.