Polynomial Division Calculator
Divide polynomials with exact long or synthetic division, use the remainder and factor theorems, and find the sum and product of the zeros.
About the Polynomial Division Calculator
Polynomial long division works like number long division: divide the leading terms, multiply back, subtract, and repeat. When the divisor is x − a, the remainder equals p(a) (the remainder theorem), and x − a is a factor exactly when p(a) = 0 (the factor theorem).
For a quadratic ax² + bx + c, the zeros α and β satisfy α + β = −b/a and αβ = c/a. For a cubic, α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a.
The formula
p(x) = d(x) × q(x) + r(x), with deg r < deg d
- Remainder theorem
- p(x) ÷ (x − a) leaves p(a)
- Factor theorem
- x − a is a factor ⟺ p(a) = 0
- Quadratic zeros
- α + β = −b/a, αβ = c/a
Worked example
(2x³ − 3x² + 4x − 5) ÷ (x − 2)
- Quotient 2x² + x + 6, remainder 7
- Check: p(2) = 16 − 12 + 8 − 5 = 7
Questions people ask
How do I enter missing powers?
Just leave them out: x^3 + 1.
Can the coefficients be fractions?
Yes, like x^2/2 + 3x. The working stays exact.