Polynomial Long Division Calculator

Write both polynomials in descending order with 0 placeholders for missing powers. Divide the leading term of the dividend by the leading term of the divisor and write the result in the quotient. Multiply the whole divisor by that result and subtract it from the dividend. Repeat with what is left until its degree is lower than the divisor's. What remains is the remainder. Enter your own polynomials above to see every round of the work.

Yes, with this one. Enter the dividend and divisor above and it shows the quotient, the remainder, every round of the division, and a check. Most scientific calculators have no polynomial division key, so on those you work it out by hand using the steps on this page. Graphing calculators with a computer algebra system can usually divide polynomials directly, but they often skip the working that a class asks you to show.

Stop when what is left is 0 or has a lower degree than the divisor. That leftover is the remainder. The rule is about degree, not about running out of terms. If you divide by x² + 2x + 1 and are left with 5x + 5, you stop, because 5x + 5 has degree 1 and the divisor has degree 2. Going further would need a power of x with a negative exponent, which is not a polynomial.

Write a 0 coefficient in every empty place before you start. For x³ + 4 that gives x³ + 0x² + 0x + 4. The zeros keep each power in its own column, so when you subtract, only terms with the same power are combined. Skipping the placeholders is one of the most common reasons a long division comes out wrong. The calculator adds them automatically and shows the padded form in its first step.

Use long division whenever the divisor is not shaped like x − c, such as x² + 2x + 1 or 3x − 5. Synthetic division only works for divisors like x − 3, though it is faster there. If your divisor is x − c, both methods give the same quotient and remainder, so use whichever your class expects. Long division is the one method that covers every case.

Multiply the divisor by the quotient, add the remainder, and compare the result with the dividend. They must match exactly. For (x³ + 2x² − 3x + 4) ÷ (x − 7), the quotient x² + 9x + 60 times x − 7 gives x³ + 2x² − 3x − 420, and adding the remainder 424 returns the original dividend. This calculator runs that check on every answer and shows it as the last step.

A remainder of zero means the divisor divides the dividend evenly, so the divisor is a factor and the dividend equals the divisor times the quotient. For example, x⁴ + 2x² + 17x − 48 divided by x + 3 leaves 0, so x + 3 is a factor. This is the idea behind finding roots. If dividing by x - c leaves a remainder of 0, then c is a root of the polynomial.

Put it over the divisor and add it to the quotient: dividend/divisor = quotient + remainder/divisor. For the first example on this page, that is x² + 9x + 60 + 424/(x − 7). If the remainder is negative, subtract it instead, as in x² − 2x + 4 − 4/(x + 2). If the remainder is 0, the answer is just the quotient.

Polynomial long division is usually introduced in Algebra 2 and used again in Precalculus, where it supports factoring higher-degree polynomials and working with rational functions. It builds on ordinary long division with whole numbers, so the main new skills are keeping powers aligned and subtracting polynomials carefully, including the signs.

The method is the same four moves every time, so it is more repetitive than hard. Most trouble comes from small slips: sign errors when subtracting, terms of different powers lined up in the same column, and forgotten zero placeholders. Working neatly in columns and checking the answer by multiplying back catches most of them. Using the calculator above to see each round can help you find where a hand attempt went off.

The usual slips are sign errors in the subtraction, misaligned columns, a missing 0 placeholder for an absent power, dividing the whole polynomial instead of only the leading terms, and stopping too early or too late. Multiplying the divisor by the quotient and adding the remainder shows right away whether an answer is wrong, and the rounds in the calculator show where it went wrong.

Yes. Then no division can happen, so the quotient is 0 and the remainder is the dividend itself. For example, x² divided by x³ + 1 gives a quotient of 0 and a remainder of x². The calculator handles this case and says so in its steps, rather than returning an error.

Yes. Coefficients can be whole numbers, decimals, or fractions written like 5/4, and the division is done in exact fractions, so there is no rounding. For example, (1/3)x³ + (5/4)x − 5/6 divided by x + 3 gives a quotient of (1/3)x² − x + 17/4 and a remainder of -163/12. Write x/2 as (1/2)x, since a slash after x is not read as division.

Divide each term of the polynomial by the monomial separately and add the results. For example, (6x³ + 9x² − 3x) ÷ 3x = 2x² + 3x − 1. That shortcut works when the divisor has one term. The long division on this page also handles it, because a monomial is just a divisor with a single term.

A fraction can only be split into partial fractions when the degree of its top is lower than the degree of its bottom. If it is not, you divide first, which gives a polynomial plus a proper fraction, and then you decompose that proper fraction. The partial fraction decomposition calculator on this site does this division step automatically before it splits the fraction.

For any polynomial A and nonzero polynomial B, there is exactly one quotient Q and remainder R such that A = B × Q + R, where R is 0 or has a lower degree than B. Long division is the method that finds Q and R. The same identity is what you use to check an answer.

When you divide a polynomial by x − c, the remainder equals the polynomial's value at x = c. Dividing x³ + 2x² − 3x + 4 by x − 7 leaves 424, and substituting 7 into the polynomial also gives 424. So the remainder can be found without dividing, and a remainder of 0 means x - c is a factor.

Use polynomial long division, which works for any two polynomials. Write both in descending order with 0 placeholders for missing powers, divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by that result, subtract, and repeat until what is left has a lower degree than the divisor. If the divisor has the form x − c, synthetic division gets the same answer with less writing. Enter both polynomials above to see the work.

Synthetic division is the shortcut, and it is the quick method, but it only works when the divisor is shaped like x − c, a linear factor with a leading coefficient of 1. It uses only the coefficients, multiplying and adding instead of writing out full terms, and it gives the same quotient and remainder as long division. For any other divisor, such as x² + 2x + 1, long division is the method to use, and this calculator handles every kind of divisor.

Programs that divide polynomials and carry out synthetic division have been written in TI-Basic for the TI-83 and TI-84 Plus, and they can be typed into the calculator. Beyond that, most classes expect you to work the division by hand. Models and software versions differ, so check your own calculator's manual before relying on a particular feature. The calculator on this page does the whole division and shows each step, which makes it a fast way to check a hand attempt.

The box method, also called the area model, lays the division out as a rectangle. The divisor forms one side, the dividend is the total area, and the quotient is the missing side, found one term at a time. You divide the first dividend term by the first divisor term, fill in the matching box, then check that the like terms in the grid add up to the dividend. It gives the same quotient and remainder as long division, only drawn as a grid.

There are five steps. First, write both polynomials in descending order, with a 0 for any missing power. Second, divide the leading term of the dividend by the leading term of the divisor to get the next quotient term. Third, multiply the whole divisor by that term. Fourth, subtract the product from the dividend. Fifth, repeat with what is left until its degree is lower than the divisor's. The last leftover is the remainder.

It is the guarantee that dividing a polynomial by any non-zero polynomial gives exactly one quotient and one remainder, written dividend = divisor × quotient + remainder. The remainder is either 0 or has a lower degree than the divisor. Long division is the method that finds them, and this calculator then checks the result by multiplying back.

Yes. Long division works for any non-zero divisor. When the divisor is not monic, the quotient terms can be fractions, because each step divides by that leading coefficient. For example, 6x³ + x² − 5x + 2 divided by 3x − 1 gives 2x² + x − 4/3 with remainder 2/3. The calculator handles the fractions exactly.

The factor theorem says x - c is a factor of a polynomial exactly when dividing by x - c leaves remainder 0. So a division that ends in 0 proves the divisor is a factor and that c is a root. Any non-zero remainder is the polynomial's value at c, so the division doubles as a test.

Pick a candidate root c, usually a factor of the constant term divided by a factor of the leading coefficient. Divide by x − c. If the remainder is 0, c is a root and the quotient is a lower-degree polynomial you can keep factoring. If not, try another candidate. Repeat until you reach a quadratic you can solve directly.

Use the Euclidean algorithm. Divide the larger-degree polynomial by the smaller and keep the remainder, then divide the previous divisor by that remainder, and repeat until the remainder is 0. The last non-zero remainder is the greatest common divisor, which you can scale to make its leading coefficient 1.