Polynomial Division Calculator (long division: quotient & remainder)

Understanding the polynomial division calculator — concept and principle

This calculator divides one polynomial by another with long division and gives the quotient, the remainder, the full result and a check. For a linear divisor x − c it also applies the remainder theorem. Use it for algebra and precalculus homework, factoring, finding slant asymptotes and simplifying rational functions.

How it is calculated

1. Write both polynomials in descending powers; missing powers count as 0.
2. Divide the leading term of what is left by the leading term of the divisor; that is the next quotient term.
3. Multiply the whole divisor by that term and subtract.
4. Repeat until the remainder's degree is lower than the divisor's.

Result: P(x) = D(x)·Q(x) + R(x), or P/D = Q + R/D. For D(x) = x − c, the remainder equals P(c).

The principle

Polynomial long division follows the same pattern as long division of numbers, with powers of x playing the role of place values. Each step removes the highest power from what is left, so the degree keeps dropping and the process must end. The division algorithm guarantees a unique quotient and remainder with deg R < deg D. The remainder theorem follows directly: if P(x) = (x − c)Q(x) + R, then putting x = c makes the first term vanish, so R = P(c). A remainder of 0 means x − c is a factor and c is a root.

Worked example

Default inputs: P(x) = 2x³ − 3x² + 4x − 5, D(x) = x − 2.

1. 2x³ ÷ x = 2x². Subtract 2x²(x − 2) = 2x³ − 4x²: left x² + 4x − 5.
2. x² ÷ x = x. Subtract x(x − 2) = x² − 2x: left 6x − 5.
3. 6x ÷ x = 6. Subtract 6(x − 2) = 6x − 12: left 7.
4. Quotient 2x² + x + 6, remainder 7.
5. Check: P(2) = 16 − 12 + 8 − 5 = 7, matching the remainder theorem. The quotient has degree 2.

How to read the result

If the remainder is 0, the divisor is a factor and you can factor P as D·Q. If not, the result Q + R/D shows the polynomial part and the leftover fraction. For rational functions, when the numerator's degree is exactly one more than the denominator's, the quotient gives the slant asymptote.

Common mistakes

• Leaving out missing powers, such as dividing x³ + 1 without writing 0x² + 0x.
• Subtracting only the first term instead of the whole product.
• Sign errors when subtracting negative terms.
• Stopping before the remainder's degree is lower than the divisor's.

Frequently asked questions

When can I use synthetic division instead?

When the divisor is linear of the form x − c. It is a shortcut for the same calculation.

What does a zero remainder mean?

The divisor divides P exactly, so it is a factor of P.

Can the quotient have fractional coefficients?

Yes, when the leading coefficient of the divisor does not divide the dividend's coefficients evenly.

Sources: OpenStax, Intermediate Algebra 2e, §5.4 Dividing Polynomials