Enter \(P(x)\) and \(D(x)\), then press Calculate.
The output will show \(P(x)=D(x)Q(x)+R(x)\) with \(\deg R<\deg D\).
Drag to pan. Mouse wheel zooms (cursor-centered). Double-click resets view. Axis numbers are shown.
Polynomial Division Tool
Math Algebra • Polynomial and Rational Functions
Frequently Asked Questions
What does polynomial division return in the Polynomial Division Tool?
It returns a quotient Q(x) and a remainder R(x) so that P(x)=D(x)Q(x)+R(x). The remainder is either 0 or has degree smaller than the divisor.
How is synthetic division different from long division?
Synthetic division is a compact table method used only when the divisor is linear, such as x-c or ax+b. Long division works for any polynomial divisor and follows repeated leading-term matching and subtraction.
Why does the result require deg R < deg D?
That condition makes the remainder smaller than the divisor, which ensures the division is complete and the quotient is uniquely determined. If deg R were at least deg D, you could keep dividing further.
How can I verify that my division is correct?
Use the verification option to compare P(x) with D(x)Q(x)+R(x). If they match, the quotient and remainder satisfy the polynomial division identity.
What does the remainder mean when dividing by x-c?
When the divisor is x-c, the remainder equals P(c), which is the Remainder Theorem. If P(c)=0, then x-c is a factor of P(x).