Divide polynomials using long division or synthetic division. See quotient, remainder, step-by-step work, and factor verification.
Polynomial division is the process of dividing one polynomial by another, producing a quotient and a remainder, much like long division of integers. This calculator supports two methods: traditional long division (works for any divisor) and synthetic division (a shortcut when dividing by a linear factor of the form x − c). Enter the coefficients of the dividend (up to degree 6) and the divisor (up to degree 4), and the calculator will display the quotient polynomial, the remainder, and a detailed step-by-step breakdown of every subtraction in the long-division process. The results are verified by computing Quotient × Divisor + Remainder to confirm it equals the original dividend. Synthetic division is automatically available when the divisor is linear and monic. The step table shows each intermediate row so you can follow the algorithm as if you were working it by hand. This tool is indispensable for factoring higher-degree polynomials, applying the Remainder Theorem, and checking roots. Use the preset buttons to explore textbook-classic division problems instantly.
Polynomial Division Calculator helps you solve polynomial division problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter your inputs once and immediately inspect Dividend, Divisor, Quotient to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Dividend = Quotient × Divisor + Remainder, where deg(Remainder) < deg(Divisor). Remainder Theorem: dividing P(x) by (x − c) gives remainder P(c). Factor Theorem: (x − c) is a factor of P(x) if and only if P(c) = 0.
Result: Dividend shown by the calculator
Using the preset "Long Division", the calculator evaluates the polynomial division setup, applies the selected algebra rules, and reports Dividend with supporting checks so you can verify each transformation.
This calculator takes the problem inputs and applies the relevant polynomial division relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Dividend, Divisor, Quotient, Remainder to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Synthetic division works only when dividing by a monic linear polynomial, i.e., (x − c). For non-linear or non-monic divisors, use long division.
The quotient is 0 and the remainder is the dividend itself. Use this as a practical reminder before finalizing the result.
When you divide P(x) by (x − c), the remainder equals P(c). This calculator verifies this automatically.
A zero remainder means the divisor divides evenly into the dividend — the divisor is a factor. Keep this note short and outcome-focused for reuse.
Yes. This calculator supports divisors up to degree 4, so quadratic, cubic, and quartic divisors are all valid.
The calculator computes Quotient × Divisor + Remainder and checks that it equals the Dividend. The verification row confirms correctness.