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Polynomial long division works just like the long division you learned with numbers — divide, multiply, subtract, bring down — but with terms instead of digits. Unlike synthetic division, this method works for divisors of any degree, not just linear ones.
How it works
At each step, the leading term of the current remainder is divided by the divisor's leading term to get the next quotient term; the divisor times that term is then subtracted from the remainder, and the process repeats on what's left.
- Enter dividend: x⁵ coefficient (optional).
- Enter dividend: x⁴ coefficient (optional).
- Enter dividend: x³ coefficient.
- Enter dividend: x² coefficient.
- Enter dividend: x coefficient.
- Enter dividend: constant.
- Enter divisor: x³ coefficient (optional).
- Enter divisor: x² coefficient (optional).
- Enter divisor: x coefficient.
- Enter divisor: constant.
- Click Calculate to see your results.
Examples
A clean division
(x³-6x²+11x-6) ÷ (x-1) = x²-5x+6 with remainder 0.
A division with a remainder
(x²+3x+5) ÷ (x+1) = x+2 with remainder 3.
Who should use it
- Algebra 2 and precalculus coursework on polynomial long division.
- Checking long division homework before a test.
Industry applications
- Mathematics education
Advantages
- Handles divisors of any degree, not just linear ones.
- Shows every divide-multiply-subtract step explicitly.
Limitations
- Limited to a dividend degree up to 5 and divisor degree up to 3.
Common mistakes to avoid
- Forgetting to include a placeholder (zero coefficient) for a missing power of x in the dividend, which misaligns every subsequent subtraction step.
Best practices
- Write out every power of x in both the dividend and divisor (using 0 for any missing terms) before starting the division, just as you'd do by hand.
Tips
- If the remainder comes out to 0, the divisor is a factor of the dividend — a useful way to confirm a suspected root or factor.