Stepwise.

Polynomial Division Calculator

Divide one polynomial by another with long division or synthetic division, showing the quotient, remainder and every step of the working.

How to use this calculator

  1. 1Choose long division for any divisor, or synthetic division when dividing by x − r.
  2. 2Enter the dividend coefficients highest power first, with a zero for every missing term.
  3. 3Enter the divisor the same way, or just the value of r for synthetic division.
  4. 4Read the quotient and remainder, and check the parsed polynomials in the notes.

How the calculation works

P(x) = D(x) · Q(x) + R(x), deg R < deg D synthetic: bring down, multiply by r, add
P(x)
The dividend — the polynomial being divided
D(x)
The divisor — the polynomial dividing it
Q(x)
The quotient, of degree deg P − deg D
R(x)
The remainder, always of lower degree than the divisor
r
In synthetic division, the root of the divisor x − r

The division algorithm for polynomials mirrors the one for integers: any P and non-zero D give a unique Q and R with the remainder smaller than the divisor. For integers "smaller" means in size; for polynomials it means lower degree.

Each step divides the current leading term by the divisor's leading term, multiplies the whole divisor by that result, and subtracts. The leading term cancels by construction, which is what guarantees the process terminates.

The remainder theorem says the remainder on dividing by x − r equals P(r). The factor theorem is the special case: the remainder is zero exactly when x − r is a factor, so a root and a linear factor are the same fact stated two ways.

Coefficients must include zeros for missing powers. Entering x³ − 1 as "1, −1" instead of "1, 0, 0, −1" silently divides a completely different polynomial.

Worked example

x⁴ − 4x³ + 7x − 2 divided by x − 3

  1. 1.The dividend needs its missing x² term: coefficients are 1, −4, 0, 7, −2.
  2. 2.x⁴ ÷ x = x³. Multiply x − 3 by x³ to get x⁴ − 3x³ and subtract, leaving −x³ + 0x² + 7x − 2.
  3. 3.−x³ ÷ x = −x². Multiply to get −x³ + 3x² and subtract, leaving −3x² + 7x − 2.
  4. 4.−3x² ÷ x = −3x. Multiply to get −3x² + 9x and subtract, leaving −2x − 2.
  5. 5.−2x ÷ x = −2. Multiply to get −2x + 6 and subtract, leaving −8.
  6. 6.Quotient x³ − x² − 3x − 2, remainder −8.

Result: x³ − x² − 3x − 2, remainder −8

A divisor that divides exactly

  1. 1.P(x) = x³ − 6x² + 11x − 6, divided by x − 1.
  2. 2.Synthetic division: bring down 1. Multiply by 1 and add to −6, giving −5.
  3. 3.Multiply −5 by 1 and add to 11, giving 6. Multiply 6 by 1 and add to −6, giving 0.
  4. 4.The final 0 is the remainder, so x − 1 is a factor and 1 is a root.
  5. 5.Remainder theorem check: P(1) = 1 − 6 + 11 − 6 = 0, confirming it.
  6. 6.The quotient x² − 5x + 6 factors further as (x − 2)(x − 3), so the roots are 1, 2 and 3.

Result: x² − 5x + 6, remainder 0

Why polynomial division is easier than it looks

Long division of numbers requires guessing how many times the divisor goes into a partial remainder, and carrying between place values. Polynomial division has neither problem. Each step is forced: divide the current leading term by the divisor's leading term, and there is exactly one answer. Nothing carries, because x² and x are separate slots that never interact.

That is why the process always terminates, and why the answer is unique. Each subtraction is designed to cancel the leading term, so the working polynomial drops at least one degree every step. Once its degree falls below the divisor's, nothing further can be extracted and what remains is the remainder.

The requirement that the remainder have lower degree than the divisor is the exact analogue of requiring an integer remainder to be smaller than the divisor. It is what makes the quotient and remainder unique rather than merely possible.

Synthetic division: the same work with the writing removed

Synthetic division is not a different algorithm. It is long division with the variables, the plus signs and the repeated leading terms stripped out, leaving only the coefficients that actually change. For a divisor of x − r the whole procedure collapses to: bring down the first coefficient, multiply by r, add to the next, repeat.

The cost of that compression is generality. It relies on the divisor being monic — leading coefficient exactly 1 — and linear. Dividing by 2x − 6 or by x² + 1 requires either adjusting for the leading coefficient or abandoning the shortcut for full long division.

This calculator therefore offers synthetic division only for divisors of the form x − r, and long division for everything else. Note the sign trap: to divide by x + 2 you enter r = −2, because x + 2 is x − (−2). Getting that backwards is the most common synthetic division error after forgetting a zero coefficient.

The remainder and factor theorems

Divide P(x) by x − r and the remainder is always P(r), the polynomial evaluated at r. This is the remainder theorem, and it falls straight out of the division statement: P(x) = (x − r)Q(x) + R. Substituting x = r kills the first term entirely, leaving P(r) = R.

The factor theorem is the case where that remainder is zero. If P(r) = 0, then x − r divides P exactly and is a factor; conversely, if x − r is a factor then r is a root. Root-finding and factoring are the same problem wearing different clothes.

Together they make synthetic division a practical root-hunting tool. Test a candidate root, and a zero remainder both confirms it and hands you the reduced polynomial to keep working on — one degree lower, and often factorable by inspection. That is how cubics and quartics are usually cracked by hand.

What this assumes, and where it stops

Assumptions

  • Coefficients are entered highest power first, including explicit zeros for missing terms.
  • Both polynomials are in a single variable with real coefficients.
  • Synthetic division assumes a monic linear divisor, x − r.

Limitations

  • Single-variable polynomials only. Multivariable division needs an ordering of terms and a different algorithm.
  • Real coefficients only — complex coefficients are not accepted.
  • Arithmetic is floating point, so coefficients that should cancel exactly can leave values like 1e−16. Results are rounded for display, and a remainder within 10⁻¹⁰ of zero is reported as exact.
  • The calculator divides; it does not factor a polynomial completely or find its roots for you.

Common questions

How do I do synthetic division?

Write the dividend's coefficients, including zeros for missing powers. Bring down the first one. Multiply it by r and add the result to the next coefficient. Repeat to the end. The last number is the remainder and the rest are the quotient's coefficients, one degree lower than the dividend. Remember that dividing by x + 2 means r = −2.

Why do I need zeros for missing terms?

Because the position of each coefficient is what identifies its power. Entering x³ − 1 as "1, −1" tells the calculator you mean x − 1, an entirely different polynomial. Written properly it is "1, 0, 0, −1" — one x³, no x², no x, minus one.

What does a remainder of zero mean?

The divisor is a factor of the dividend, and the division is exact. If you divided by x − r, a zero remainder also means r is a root of the polynomial. That is the factor theorem, and it is why synthetic division is the standard way to test candidate roots by hand.

When should I use long division instead of synthetic division?

Whenever the divisor is not of the form x − r. Synthetic division needs a linear divisor with a leading coefficient of 1; anything quadratic or higher, or with a leading coefficient like 2x − 6, requires long division. Long division always works, so it is the safe default.

Sources

Formula and content last reviewed on .

Results are estimates for information only, not professional advice.

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