This polynomial division calculator performs long division of a polynomial (up to degree 4) by a quadratic divisor, returning the quotient and remainder: P(x) = Q(x)·D(x) + R(x), with the remainder of lower degree than the divisor. It is the general division machine behind slant asymptotes, improper-fraction cleanup for partial fractions, and the factor theorem’s heavy lifting.

Dividend P(x) = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀

Divisor D(x) = b₂x² + b₁x + b₀

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Division With a Remainder Theorem Attached

Polynomial long division mirrors integer long division move for move: divide leading terms, multiply back, subtract, bring down, repeat — until what's left (the remainder) has degree below the divisor's. The result is an identity, P = Q·D + R, that can be checked by multiplying out, and the division-algorithm theorem guarantees the quotient and remainder are unique: there's exactly one right answer, which makes polynomial division the rare algorithm whose output audits itself. The defaults demonstrate the standard cubic-by-quadratic case (quotient 2x + 1, remainder −4x − 2 — verified below by recombination). Special cases layer on top: a zero remainder certifies D as a factor (the factor theorem's computational face), and dividing by a linear (x − r) collapses the whole algorithm into synthetic division — whose remainder is exactly P(r), the remainder theorem — a faster lane with its own page here, which is why this calculator takes the quadratic divisor that synthetic division can't handle.

Downstream, division is infrastructure: it produces slant asymptotes (the quotient of a proper division is the line), it performs the “make it proper” first step partial fractions require, it deflates polynomials after the rational root theorem finds a root, and it's the Euclidean algorithm's first cousin — gcds of polynomials, hence simplifying rational expressions in theory and in computer algebra, run on repeated division.

Worked Example (Verified)

  • (2x³ + 5x² − 4x − 3) ÷ (x² + 2x − 1) (defaults): first term 2x — multiply back 2x³ + 4x² − 2x, subtract → x² − 2x − 3. Next term +1 — multiply back x² + 2x − 1, subtract → remainder −4x − 2. Quotient 2x + 1. Audit: (2x + 1)(x² + 2x − 1) + (−4x − 2) = (2x³ + 5x² + 0x − 1) + (−4x − 2) = 2x³ + 5x² − 4x − 3 ✔.
  • Exact factor case: (x³ − 6x² + 11x − 6) ÷ (x² − 5x + 6): quotient x − 1, remainder 0 — because the divisor is (x − 2)(x − 3) and the dividend (x − 1)(x − 2)(x − 3); zero remainder is factorization announcing itself.
  • Quartic reach: the widget accepts degree-4 dividends (quotient quadratic) — e.g. (x⁴ − 16) ÷ (x² + 4) = x² − 4 exactly, the difference-of-squares pattern surfacing as zero-remainder division.

Division Discipline

  • Write missing terms with 0 coefficients (x³ + 1 becomes x³ + 0x² + 0x + 1) — skipped placeholders are the leading cause of column misalignment and wrong subtractions; the widget's five explicit fields enforce this for you.
  • Subtract the whole product row, signs flipped — most long-division errors are a single un-flipped sign in the subtraction line.
  • Stop at remainder degree < divisor degree; continuing manufactures negative-degree nonsense, and stopping early leaves a fake remainder.
  • Report answers as the identity (quotient + remainder/divisor for rational functions), not a decimal — the remainder is exact information, not error.
  • Audit by multiplying back on any high-stakes division: uniqueness of Q and R means one recombination proves correctness completely.

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Frequently Asked Questions

How do you divide polynomials by long division?

Align both polynomials in descending degree with zero placeholders for missing terms. Divide the leading term of the dividend by the divisor's leading term (that's the quotient's next term), multiply the whole divisor by it, subtract, and repeat on the result until the remainder's degree is below the divisor's. Our default: (2x³ + 5x² − 4x − 3) ÷ (x² + 2x − 1) = 2x + 1 with remainder −4x − 2 — this calculator performs the algorithm and states the identity.

What are the quotient and remainder?

The unique polynomials Q and R with P = Q·D + R and deg R < deg D. Uniqueness is a theorem (the division algorithm), so Q and R are THE answer, checkable by multiplying out. Written as a rational function, P/D = Q + R/D — the split into a polynomial part and a proper fraction that partial fractions and slant-asymptote analysis both consume.

How is this different from synthetic division?

Synthetic division is the streamlined special case for linear divisors (x − r): coefficients only, no x's, much faster by hand — and it has its own dedicated page on this site. It cannot divide by a quadratic; polynomial long division (this page) handles any divisor. Rule of thumb: linear divisor → synthetic; anything fatter → long division. The remainder theorem (remainder = P(r)) also belongs to the synthetic world.

What does a remainder of zero mean?

The divisor divides the dividend exactly — D is a factor of P (factor theorem, generalized). (x⁴ − 16) ÷ (x² + 4) = x² − 4 remainder 0 because x⁴ − 16 = (x² − 4)(x² + 4). Zero remainders are how factorizations are discovered and verified: the rational root theorem proposes linear factors, division confirms them, this page confirms the quadratic groupings.

When is the quotient a slant asymptote?

When the division comes from a rational function whose numerator outdegrees its denominator by one — the quotient is then linear, and that line is the slant (oblique) asymptote: the graph of P/D approaches Q(x) because the remainder term R/D → 0. Example from our Asymptotes page: (x² + 3x + 1)/(x + 1) = (x + 2) + (−1)/(x + 1), slant line y = x + 2. Division by a quadratic, as here, yields the same idea with polynomial end-behavior.

Why write missing terms with 0 coefficients?

Because each subtraction must cancel degree-by-degree; a missing x term silently shifts every subsequent column and corrupts the arithmetic — the single most common long-division error. Placeholders keep the degrees aligned (x³ + 1 is x³ + 0x² + 0x + 1). The widget enforces it structurally: every degree has its own input field, zeros included.

Can the divisor have a leading coefficient other than 1?

Yes — the algorithm divides leading terms, so a b₂ of 2 or −3 produces fractional quotient coefficients, which are exact and fine. (Integer-coefficient coursework often factors the divisor's leading coefficient aside first to keep arithmetic clean, but that is cosmetics, not mathematics.) The widget accepts any nonzero b₂; only b₂ = 0 is refused, because then the divisor isn't quadratic and the problem belongs to synthetic division's lane.

Is this polynomial division calculator free?

Yes — free, instant, no sign-up. Enter a dividend to degree 4 and a quadratic divisor for the quotient, remainder and the verification identity.