➗

Polynomial Long Division Calculator

Q: When do I stop dividing polynomials?

When the degree of the remainder is smaller than the degree of the divisor. Dividing by a quadratic, you stop once the remainder is linear or a constant.

Q: What if a power of x is missing?

Treat it as a 0 coefficient. x³ − 8 is x³ + 0x² + 0x − 8; the calculator inserts those placeholders automatically.

Q: How do I write the answer with a remainder?

As Q(x) + R(x)/D(x). Here: 2x − 1 + (−x + 5)/(x² + 2x − 1).

Q: When should I use synthetic division instead?

Only when the divisor is linear, x − c. For anything else, such as x² + 1 or 2x² − x + 3, use long division.

Divide one polynomial by another of any degree with long division: quotient, remainder, every divide-multiply-subtract step, and the check D(x)·Q(x) + R(x) = P(x).

Dividend P(x)

Divisor D(x)

Loading calculator…

How it works

P(x) = D(x) · Q(x) + R(x), with degree R < degree D · each step: next quotient term = (leading term of remainder) ÷ (leading term of D)

Polynomial long division works like numeric long division. Divide the leading term of what is left by the leading term of the divisor to get the next term of the quotient, multiply the whole divisor by that term, subtract, and repeat until the remainder has a lower degree than the divisor. Unlike synthetic division it works for divisors of any degree. Exact fractions are kept throughout.

Worked example

(2x³ + 3x² − 5x + 6) ÷ (x² + 2x − 1): 2x³ ÷ x² = 2x; 2x · (x² + 2x − 1) = 2x³ + 4x² − 2x; subtracting leaves −x² − 3x + 6. Then −x² ÷ x² = −1; −1 · (x² + 2x − 1) = −x² − 2x + 1; subtracting leaves −x + 5. Quotient 2x − 1, remainder −x + 5.

Sources: OpenStax College Algebra 2e — 5.4 Dividing Polynomials · Paul’s Online Math Notes — Dividing Polynomials

Frequently asked questions

When do I stop dividing polynomials?

When the degree of the remainder is smaller than the degree of the divisor. Dividing by a quadratic, you stop once the remainder is linear or a constant.

What if a power of x is missing?

Treat it as a 0 coefficient. x³ − 8 is x³ + 0x² + 0x − 8; the calculator inserts those placeholders automatically.

How do I write the answer with a remainder?

As Q(x) + R(x)/D(x). Here: 2x − 1 + (−x + 5)/(x² + 2x − 1).

When should I use synthetic division instead?

Only when the divisor is linear, x − c. For anything else, such as x² + 1 or 2x² − x + 3, use long division.

Related calculators

Synthetic Division Calculator

Divide a polynomial by x − r with synthetic division: see the full tableau, the quotient, the remainder, the remainder theorem check and whether x − r is a factor.

Polynomial Calculator

Add, subtract, multiply or divide two polynomials. Type them as expressions like x^2 − 3x + 2 or (x − 1)(x + 2), or as coefficient lists, and get the simplified result, quotient and remainder, degree, coefficients, a value check and the long-division steps.

Partial Fraction Decomposition Calculator

Decompose a rational function N(x)/D(x) into partial fractions: factor the denominator, set up A/(x − r) terms (repeated and quadratic factors too), solve for the constants exactly.

Rational Root Theorem Calculator

List every possible rational root ±p/q of a polynomial with integer coefficients, test each candidate, and get the actual rational zeros with their multiplicity and the factored form.

This calculator is for educational purposes. Double-check important results before acting on them.