What does the polynomial division calculator do?

The polynomial division calculator divides a polynomial P(x) by a polynomial D(x) and returns the quotient, the remainder, and the long-division steps.

It is designed for the general case, not only for divisors of the form x−r. The result includes a verification identity so you can check that divisor × quotient + remainder reconstructs the original dividend.

How polynomial long division works

Polynomial long division repeatedly compares leading terms. Each new quotient term is chosen so that the leading term of the current remainder is cancelled.

1. Divide the leading term of the current dividend by the leading term of the divisor.
2. Multiply the whole divisor by that quotient term.
3. Subtract the product from the current dividend.
4. Repeat until the remainder has lower degree than the divisor.

This mirrors ordinary long division, but the place values are powers of x instead of digits.

What inputs are supported?

The calculator supports single-variable polynomial text inputs for P(x) and D(x). The divisor must have degree at least 1.

• The divisor may be non-monic; exact fractional quotient terms are allowed.
• A dividend with lower degree than the divisor is valid.
• Free-text general algebra, parentheses as full expressions, named functions, multivariable terms, and non-polynomial expressions are outside the calculator scope.
• The result is a polynomial quotient and a polynomial remainder.

How to read quotient and remainder

The quotient Q(x) is the polynomial part of the division. The remainder R(x) is what is left after no higher-degree cancellation is possible.

A zero remainder means the division is exact. A nonzero remainder does not make the calculation wrong; it simply means P(x) is not evenly divisible by D(x).

Worked example of a polynomial division

For P(x)=2x^3+3x^2−5x+4 and D(x)=x−2, the first quotient term is 2x^2 because 2x^3 divided by x equals 2x^2.

1. Multiply x−2 by 2x^2 and subtract the product.
2. Use the new leading term to choose the next quotient term.
3. Continue multiplying and subtracting until the remaining expression has degree 0.
4. Check the answer with D(x) × quotient + remainder.

The calculator shows these actions in a step table so you can see where each quotient term and each new remainder comes from.

What if the dividend degree is lower?

If the degree of P(x) is lower than the degree of D(x), the division is already finished.

The quotient is 0 and the remainder is P(x). For example, dividing x+1 by x^2+3 gives quotient 0 and remainder x+1.

Polynomial division vs synthetic division

Polynomial division is the general method. Synthetic division is a shortcut for monic linear divisors such as x−r.

• Use synthetic division for x−3 when you want the compact table.
• Use polynomial division for x^2+1, 2x−3, or any divisor that is not the classic monic linear case.
• Use polynomial division when you need the full quotient-remainder identity.

The two calculators are related, but they serve different user jobs.

Common mistakes and limitations

• Forgetting missing zero-coefficient terms.
• Stopping before the remainder degree is lower than the divisor degree.
• Treating a nonzero remainder as an error.
• Rounding fractional quotient coefficients when the divisor is non-monic.
• Using this tool for multivariable or non-polynomial expressions.