Polynomial Division Calculator

Introduction

Polynomial division splits one polynomial by another. It is the polynomial twin of the long division you learned with whole numbers: the cast is the same (dividend, divisor, quotient, remainder), the loop is the same (divide the leading terms, multiply, subtract, bring the next term down), and the guarantee is the same (the remainder is "smaller" than the divisor). What changes is the meaning of smaller. For integers the remainder must be less than the divisor; for polynomials the remainder must have a strictly lower degree than the divisor. [openstax-dividing-polynomials]

This calculator performs that division from a coefficient list. You type the coefficients of the dividend, type the coefficients of the divisor, and press Divide. The tool returns four results: the quotient polynomial, the remainder polynomial, the degree of the quotient, and — when the remainder collapses to a constant — a plain Yes or No stating whether the divisor divides the dividend evenly. [wikipedia-poly-division]

A word about scope, because the two tools sound alike from the outside. The Long Division Calculator divides numbers: 4,876 by 12 gives 406 with remainder 4. This calculator divides polynomials: x^3 - 1 by x - 1 gives x^2 + x + 1 with remainder 0. The arithmetic underneath is closely related — polynomial long division is integer long division with degree playing the role of place value — but the inputs, outputs, and uses are different. If your dividend is a plain number, use the number tool. If your dividend is an expression such as 2x^3 - 3x^2 + 4x - 5, you are in the right place.

The procedure earns its place in the curriculum because it does three jobs at once. First, it reduces degree: dividing a cubic by a linear factor leaves a quadratic, which you can then hand to the Quadratic Equation Calculator or the Quadratic Formula Calculator. Second, it tests factors: if the division comes out even, the divisor really is a factor, and factor theorem guarantees nothing is lost in the process. Third, it simplifies expressions: rational functions, limits, integrals requiring partial fractions, and asymptotic analysis all start by dividing numerator by denominator.

Polynomial division also sits at the centre of a small web of related tools on this site. The Factor Calculator factors integers — a different problem with a different algorithm, but the same idea of breaking an object into irreducibles, and the two are often used together when a problem mixes integer and polynomial structure. The Expanded Form Calculator rewrites products as sums, which is exactly how you verify a division: multiply the quotient by the divisor, add the remainder, and expand. Together these three tools let you move freely between factored form, expanded form, and quotient-remainder form. [khan-poly-division]

One clarification that saves confusion later: "dividing polynomials" on this page always means exact symbolic division, not numerical evaluation at a point. The calculator does not plug in a value of x; it produces a formula for the quotient and a formula (often just a number) for the remainder. Evaluation enters the story only through the remainder theorem, which is the bridge between the two viewpoints and is covered in full in the formula section below.

How to Use

The interface takes two text fields and one button:

1. Dividend Coefficients — the coefficients of the polynomial you are dividing, written in descending order of degree and separated by commas. For x^2 - 3x + 2 you type 1, -3, 2.
2. Divisor Coefficients — the coefficients of the polynomial you are dividing by, same convention, same comma separators. For x - 1 you type 1, -1.
3. Press Divide. The results panel shows the quotient polynomial, the remainder polynomial, the degree of the quotient, and — whenever the remainder is a constant — the Yes/No flag Divisible Without Remainder.

Two conventions make or break the input. Missing terms must be written as 0. The polynomial x^3 - 1 has no x^2 term and no x term, so its coefficient list is 1, 0, 0, -1, not 1, -1. Forgetting a placeholder zero silently changes the degree of the polynomial, and every result after that will be wrong but plausible — the worst kind of error. Signs are part of the coefficient: subtracting 3x means the coefficient is -3, not 3.

Example 1 — a division that comes out even. Divide x^2 - 3x + 2 by x - 1. Enter 1, -3, 2 as the dividend and 1, -1 as the divisor. The quotient is x - 2, the remainder is 0, the degree of the quotient is 1, and the divisibility flag reads Yes. Because both roots are visible after the division, this is the classic warm-up: dividing by x - 1 peeled off the factor, and x - 2 was left standing.

Example 2 — a remainder that will not go away. Divide x^2 + 2x + 3 by x + 1. Enter 1, 2, 3 and 1, 1. The quotient is x + 1, the remainder is 2, the degree of the quotient is 1, and the flag reads No. The remainder being a nonzero constant tells you the divisor is not a factor, and — by the remainder theorem — that the dividend evaluated at x = -1 equals 2.

Example 3 — a cubic with missing terms. Divide x^3 - 1 by x - 1. Enter 1, 0, 0, -1 for the dividend (note the two zeros) and 1, -1 for the divisor. The quotient is x^2 + x + 1, the remainder is 0, and the degree of the quotient is 2. This is the difference-of-cubes factorisation in action, produced by a single division.

Example 4 — a divisor that is not linear. Divide x^4 - 1 by x^2 - 1. Enter 1, 0, 0, 0, -1 and 1, 0, -1. The quotient is x^2 + 1, the remainder is 0, and the degree of the quotient is 2. Nothing in the algorithm requires the divisor to be linear; only the shorthand method called synthetic division does, and that restriction is explained below.

Reading the results takes a moment's care in the general case. When the divisor has degree greater than one, the remainder can itself be a polynomial of degree one or higher — for example, dividing x^2 - 1 by x^2 + 1 leaves a constant remainder of -2 but dividing x^3 by x^2 + 1 leaves -x, which has degree 1. In those cases the "divisible without remainder" question is not a yes/no matter in the useful sense, so the flag is withheld rather than reported as a misleading No.

The Formula

Everything on this page follows from a single identity. For any dividend p(x) and any nonzero divisor d(x) there exist unique polynomials q(x) and r(x) with

The degree condition is the whole engine of the algorithm: a remainder of degree lower than the divisor cannot be divided any further by it, so the process must stop. This mirrors integer division, where the remainder must be smaller than the divisor, and it is what makes the process both finite and unique. [openstax-dividing-polynomials]

Reference Table: Dividend, Divisor, Quotient and Remainder

Two patterns are worth extracting from this table. First, the degree column obeys the rule without exception: it is always the dividend's degree minus the divisor's degree. A cubic divided by a linear divisor yields degree 2; a quadratic divided by a quadratic yields degree 0, which is why the last row's quotient is the bare constant 1. Second, the remainder is zero in four of the seven rows, and every one of those rows is a factorisation in disguise — x - 1 divides x^2 - 3x + 2, x + 3 divides x^3 + 2x^2 - 5x - 6, x^2 - 1 divides x^4 - 1. Rows with nonzero remainders are equally informative: they are counterexamples that prove a candidate factor is not actually a factor. [purplemath-polydiv3]

The remainder theorem turns the same rows into a table of function values. For p(x) = x^3 + 2x^2 - 5x - 6, the divisor x - r and the remainder p(r) are two views of one number:

The two zeros at r = -3 and r = -1 announce two linear factors, and dividing them out leaves x - 2, so the polynomial factors completely as (x + 3)(x + 1)(x - 2). Note the sign flip built into the correspondence: the divisor is x + 3 while r is -3, because the divisor must be written as x - r. Misreading that sign is the most common error when switching between the two notations. [mathsisfun-remainder-factor]

Practical Tips

Write every missing term as 0. The single most common input mistake is compressing x^3 - 1 into 1, -1. That list describes x - 1, a completely different polynomial, and the calculator cannot detect the intent behind it. Before typing, write the polynomial with all powers present — x^3 + 0x^2 + 0x - 1 — then read off the coefficients.

Flip every sign when you subtract. Subtracting a product of three terms means changing three signs. A reliable habit is to write the product in parentheses with a minus in front, then distribute the minus before combining anything. Most long division errors are sign errors on this single line.

Use the remainder theorem to check a division in seconds. If you divided by x - r, evaluate the dividend at r. That number must equal the remainder, whatever the quotient says. For a divisor of the form 2x - 1, evaluate at x = 1/2 instead — the remainder is unchanged. [mathsisfun-remainder-factor]

Prefer synthetic division for linear divisors, long division for everything else. Synthetic division is faster and less error-prone, but only for divisors of the form x - r. For divisors such as x^2 + 3 or x^2 - x + 1, use the full long division layout, and remember that the remainder may then be a polynomial rather than a number. [purplemath-synthdiv]

Treat a zero remainder as a factoring win, then keep going. When the flag reads Yes, the divisor is a factor. Divide the quotient by another suspected factor and reduce the degree again. Two successful divisions turn a quartic into a quadratic, at which point the Quadratic Equation Calculator finds the remaining roots directly. [purplemath-factrthm]

Sanity-check the degree before trusting the numbers. The quotient's degree is fixed by the inputs alone. If the calculator reports degree 3 for a cubic divided by a linear divisor, something about the input was not what you thought — most likely a dropped zero changed the dividend's degree. [wikipedia-poly-division]

Limitations

Coefficients are typed as text, so structure is on you. The tool cannot know that you meant a missing term; it only sees the numbers you send. A dropped zero produces a different polynomial and confidently wrong results.

Linear-divisor shortcuts are not universal. Synthetic division in its compact form applies only to divisors of the shape x - r. For non-monic divisors such as 2x - 1, the shorthand produces the quotient for x - 1/2, which must then be scaled down by the leading coefficient of the divisor. [purplemath-synthdiv]

The divisibility flag is only meaningful for constant remainders. When the divisor has degree two or more, the remainder may itself have degree one or higher — dividing x^3 by x^2 + 1 leaves -x. A yes/no verdict makes no sense there, so the flag is shown only when the remainder is a constant.

No step-by-step trace inside the tool. The calculator returns final results, not the intermediate rows of the algorithm. The worked layouts on this page exist precisely to fill that gap; combine them with the tool's quotient and remainder to reconstruct any step you need.

Real coefficients only, and exactness is fragile. The calculation is symbolic, but coefficients entered as decimals still go through floating-point arithmetic, so a division that should leave remainder 0 can come back as a tiny non-zero value. Use integer coefficients when you expect an exact result.

Finding roots is not the same as proving them. A successful division confirms the factor you supplied; it does not discover irrational or complex factors. A polynomial such as x^2 + 1 has no real root and no linear factor over the reals, and no amount of dividing by x - r will change that.

Frequently Asked Questions

❓ How is this different from the long division calculator?

✅ The long division calculator divides numbers, for example 4876 by 12. This calculator divides polynomials, for example x^3 - 1 by x - 1. The steps look similar because polynomial division is integer long division with degree in place of place value, but the inputs and outputs are different.

❓ What exactly do I type into the coefficient fields?

✅ The coefficients of the polynomial in descending order of degree, separated by commas. For x^2 - 3x + 2 you type 1, -3, 2. For the divisor x - 1 you type 1, -1. The sign of each coefficient is part of the number, so subtracting 3x means entering -3.

❓ How do I enter a polynomial with missing terms like x^3 - 1?

✅ Write a zero for every missing power: x^3 - 1 becomes 1, 0, 0, -1. Adding the placeholders keeps each coefficient aligned with its power, and it keeps the degree correct. Dropping the zeros changes the polynomial and produces wrong results.

❓ What does Divisible Without Remainder = Yes mean?

✅ It means the remainder is exactly zero, so the divisor is a factor of the dividend. For x^2 - 3x + 2 divided by x - 1 the quotient is x - 2 and the remainder is 0, so the answer is Yes. The flag appears only when the remainder is a constant; for higher degree divisors it is omitted.

❓ What is synthetic division and when can I use it?

✅ It is long division with the variable removed, using just the coefficients and the number r from a divisor of the form x - r. Because only coefficients take part in the arithmetic, the letters were never doing any work. It applies only to linear divisors of that shape; for anything else use the full long division layout.

❓ Why is synthetic division a valid shortcut?

✅ Every step of long division divides, multiplies and subtracts coefficients, and the powers of x act only as position markers that never affect the arithmetic. Suppressing them changes nothing about the computation, which is why synthetic division and long division always produce the same quotient and remainder.

❓ What is the remainder theorem?

✅ It says that the remainder from dividing a polynomial p(x) by x - r equals p(r), the value of the polynomial at r. Dividing 2x^3 - 3x^2 + 4x - 5 by x - 2 leaves remainder 7, and evaluating at x = 2 also gives 7. It turns a division problem into a quick substitution.

❓ What is the factor theorem?

✅ It says that x - r is a factor of p(x) exactly when p(r) = 0. So you can test a candidate factor by evaluating the polynomial instead of dividing it. When the test succeeds, dividing by the confirmed factor reduces the degree and leaves a smaller polynomial to factor further.

❓ How do I check whether a division is correct?

✅ Multiply the quotient by the divisor, add the remainder, and expand. The result must equal the original dividend. This is the identity p(x) = d(x) times q(x) plus r(x), and it works for every divisor, linear or not. You can also evaluate the dividend at r to check the remainder alone.

❓ Can the divisor be a quadratic or higher?

✅ Yes. The algorithm makes no assumption about the divisor's degree. Dividing x^4 - 1 by x^2 - 1 gives the quotient x^2 + 1 with remainder 0 and degree 2. Only the compact synthetic division shortcut is limited to linear divisors.

❓ Why is the degree of the quotient always the dividend's degree minus the divisor's degree?

✅ Each round of the algorithm produces one quotient term and lowers the degree of the working remainder by one. The rounds continue until the remainder's degree falls below the divisor's, so the number of quotient terms is exactly the degree gap between the two polynomials.

❓ What happens when the divisor is not monic, like 2x - 1?

✅ Long division handles it directly, with the same divide-multiply-subtract loop. For synthetic division you use r equal to the value that makes the divisor zero, which is 1/2, and then divide the resulting quotient by 2. The remainder itself is unaffected by that rescaling.