Polynomial Long Division Calculator

Q: Does this calculator show polynomial long division steps?

Yes. Each stage shows the leading-term division, divisor multiplication and subtraction, followed by exact reconstruction.

Q: Can I omit a missing power such as x²?

Yes. Missing powers are treated as zero coefficients. Terms can be entered out of order and like powers are combined.

Q: Can the quotient contain fractions?

Yes. A divisor with a leading coefficient other than 1 can produce exact fractional quotient and remainder coefficients. Inputs still require integer coefficients.

Q: What if the divisor has higher degree?

The quotient is zero and the remainder is the original dividend.

Q: Is division by a constant supported?

Yes, provided the constant divisor is nonzero. The zero polynomial is never a valid divisor.

Q: Is this synthetic division or polynomial factoring?

No. It performs polynomial long division. Those are separate procedures, and the factoring calculator remains unchanged.

About this polynomial long division calculator

This polynomial long division calculator divides one expanded polynomial in x by another. It reports the exact quotient and remainder and shows the repeated divide, multiply and subtract stages. A final reconstruction compares every coefficient with the original dividend.

It is not the same as the existing Long Division Calculator, which divides whole numbers, or the Polynomial Factoring Calculator, which finds supported factors. Polynomial division can return a nonzero remainder even when a factoring task would be inappropriate.

How to divide polynomials with this calculator

Enter the dividend in expanded form using the variable x.
Enter the nonzero divisor in the same notation. For example, x^3 − 2x + 4 divided by x − 1.
Use integer coefficients and ^ for powers. Terms may be entered out of order; like powers are combined automatically.
Calculate and read the quotient, remainder and staged working. Missing powers are treated as having zero coefficients.

Editing either input clears the old answer. Reset empties both fields. Copy, Download and Print include all working. The form does not accept a single quotient expression, an equation, parentheses or fractional input coefficients; enter each expanded polynomial separately.

Affiliate disclosure: As an Amazon Associate I earn from qualifying purchases.

For a separate arithmetic check, you may browse scientific calculators (paid link). Check the model’s supported functions; ordinary scientific models do not necessarily solve symbolic absolute-value equations or perform polynomial long division. Check supported functions and use this page’s exact working for the method. No purchase is required.

Divide, multiply, subtract and repeat

Each stage divides the leading remainder term by the leading divisor term to choose the next quotient term. It multiplies the entire divisor by that term, subtracts the product and repeats while the remainder still has a sufficiently high degree.

The process stops when the remainder is zero or its degree is lower than the divisor’s degree. Non-monic divisors can produce fractional quotient coefficients even when every entered coefficient is an integer. Those fractions are kept exact instead of being rounded. Use the Fraction Calculator to inspect a coefficient division separately.

This page implements polynomial long division. It does not offer synthetic division, factor searching or a symbolic simplifier for arbitrary expressions.

Worked polynomial division examples

Divide x^3 − 2x + 4 by x − 1. The quotient is x^2 + x − 1 and the remainder is 3. Reconstructing (x − 1)(x^2 + x − 1) + 3 gives the entered dividend exactly.

Divide x^2 + 1 by 2x + 1. The quotient is (1/2)x − 1/4 and the remainder is 5/4. Fractional coefficients are a normal outcome, not an input or rounding error.

If the divisor degree is higher than the dividend degree, the quotient is zero and the remainder is the original dividend. A zero dividend divided by a nonzero polynomial gives zero quotient and remainder. A constant nonzero divisor is also supported. For polynomial roots of degree two, use the Quadratic Formula Calculator rather than treating the quotient as a root list.

Quotient, remainder and exact verification

The reported polynomial identity is P(x) = D(x)Q(x) + R(x). The calculator multiplies the displayed divisor and quotient, adds the remainder and compares every coefficient with the normalized dividend. It also checks that a nonzero remainder has a smaller degree than the divisor.

As a quotient of functions, P(x)/D(x) = Q(x) + R(x)/D(x) only where D(x) is nonzero. A successful polynomial identity does not make the original function defined at a zero of its denominator. The tool does not find all excluded x values or cancel factors on your behalf.

A numerical check at selected x values can be useful for learning but is not a replacement for the coefficient-by-coefficient verification. The Scientific Calculator can assist with such substitutions.

Polynomial notation and supported limits

Each polynomial accepts up to 180 characters and 40 signed terms. Powers are nonnegative integers from 0 through 10. Individual coefficients and coefficients after combining like terms must be within −10,000 to 10,000. Use lowercase x, integer constants, +, −, optional multiplication * and ^ for powers.

Decimals, fractions, negative powers, other variables, equations, commas and unexpanded parentheses are rejected. Output fractions are exact but are not accepted as new input under this version’s integer-coefficient parser. The zero polynomial cannot be used as the divisor.

Constant and zero cases are labelled explicitly in the working. This page performs one polynomial division, not general equation solving or rational-function domain analysis. For a system of linear equations, use the System of Equations Calculator. Choosing the appropriate mathematical task prevents a correct division result from being mistaken for an unrelated solution.