Polynomial Division Calculator — Long & Synthetic (Free)

A polynomial division calculator divides one polynomial by another and gives the quotient and remainder. For (x³ − 2x² − 4) ÷ (x − 3), write the missing x term as 0x, then divide, multiply, subtract and bring down: the quotient is x² + x + 3 and the remainder is 5, so the answer is x² + x + 3 + 5/(x − 3).

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How to use the polynomial division calculator

1. Type the dividend and the divisor, for example x^3 – 2x^2 – 4 and x – 3. You can also switch to Coefficients and type 1, -2, 0, -4 and 1, -3 (highest power first, 0 for a missing power).
2. Pick a method: Long division works for any divisor, Synthetic is the shortcut for a linear divisor such as x − 3 or 2x + 1, and Box method shows the area model.
3. Read the quotient, the remainder and the answer written as quotient + remainder/divisor. For a linear divisor you also get the remainder theorem check.
4. Follow the layout (the quotient sits on top, aligned with like powers) and open Show steps for every divide, multiply and subtract.

Dividend, divisor, quotient and remainder

Polynomial division works like number division. The polynomial being divided is the dividend P(x), the one you divide by is the divisor D(x), the result is the quotient Q(x), and what is left over is the remainder R(x). They always satisfy the division algorithm:

P(x) = D(x) · Q(x) + R(x), where the degree of R is less than the degree of D.

So the answer can be written as P(x)/D(x) = Q(x) + R(x)/D(x). If R(x) = 0, the divisor is a factor of the dividend.

Polynomial long division step by step

Write both polynomials in descending powers and put a 0 placeholder for every missing power. Then repeat four moves until the remainder has a lower degree than the divisor:

1. Divide the leading term of what is left by the leading term of the divisor. Write the result on top.
2. Multiply that term by the whole divisor.
3. Subtract the product (change every sign, then add).
4. Bring down the next term.

Example 1: (x³ − 2x² − 4) ÷ (x − 3)

Rewrite the dividend as x³ − 2x² + 0x − 4. First, x³ ÷ x = x²; multiply x²(x − 3) = x³ − 3x²; subtract to get x², and bring down 0x. Next, x² ÷ x = x; x(x − 3) = x² − 3x; subtract to get 3x, and bring down −4. Last, 3x ÷ x = 3; 3(x − 3) = 3x − 9; subtract to get 5. The quotient is x² + x + 3 and the remainder is 5.

Check: (x − 3)(x² + x + 3) + 5 = x³ − 2x² − 9 + 5 = x³ − 2x² − 4.

Example 2: dividing by a quadratic

(2x⁴ − 3x³ + 5x − 1) ÷ (x² + x − 2). The dividend needs a 0x² placeholder. Three rounds give the quotient 2x² − 5x + 9, and the remainder is −14x + 17, which has degree 1, less than the divisor’s degree 2. The answer is 2x² − 5x + 9 + (−14x + 17)/(x² + x − 2).

Synthetic division

When the divisor is linear, x − r, synthetic division does the same work with numbers only. Write r in a box and the dividend’s coefficients beside it (0 for missing powers). Bring the first coefficient down, multiply it by r, add it to the next coefficient, and repeat. The last number is the remainder; the others are the quotient’s coefficients, one degree lower than the dividend.

Example 3: (x³ − 1) ÷ (x − 1)

Here r = 1 and the coefficients are 1, 0, 0, −1. The bottom row is 1, 1, 1, 0. The quotient is x² + x + 1 and the remainder is 0, so x − 1 is a factor: x³ − 1 = (x − 1)(x² + x + 1).

Example 4: a divisor like 2x − 1

For (6x³ + 5x² − 7) ÷ (2x − 1), use r = 1/2 because 2x − 1 = 0 when x = 1/2. Synthetic division on 6, 5, 0, −7 gives 6, 8, 4 and a remainder of −5. Since 2x − 1 = 2(x − 1/2), divide 6, 8, 4 by 2: the quotient is 3x² + 4x + 2 and the remainder stays −5. Many students forget this last division. The dedicated synthetic division calculator goes deeper into this method.

The box method

The box (area) method sets the divisor’s terms down the side of a grid and fills it so the cells add up to the dividend. For (x² + 5x + 6) ÷ (x + 2), the top-left cell must be x², so the first column heading is x² ÷ x = x; that column holds x² and 2x. The remaining 3x needs a second column headed 3, holding 3x and 6. The cells add to x² + 5x + 6 exactly, so the quotient is x + 3 with remainder 0.

Remainder theorem and factor theorem

When you divide P(x) by x − r, the remainder equals P(r). For P(x) = x² − 2x − 4 and divisor x − 3, P(3) = 9 − 6 − 4 = −1, which is exactly the remainder from long division. If P(r) = 0, the remainder is 0 and x − r is a factor. This is how synthetic division helps find roots and factor polynomials of degree 3 and higher.

Which division method should you use?

Common mistakes in polynomial division

• Skipping placeholders. x³ − 4 must be written x³ + 0x² + 0x − 4, or the columns slide out of line.
• Subtracting only the first term. Change the sign of every term in the product before adding.
• Using the wrong sign for r. Dividing by x + 2 means r = −2 in synthetic division.
• Stopping too early or too late. Stop exactly when the remainder’s degree is less than the divisor’s degree.
• Writing the remainder alone. The answer is Q(x) + R(x)/D(x), not “Q(x) R 5”, unless your teacher asks for that form.

Checking homework with the calculator

Divide by hand first, then enter the same polynomials here. If your quotient differs, compare your work with the layout row by row; the first row that does not match shows the mistake, usually a sign in a subtraction or a missing placeholder. Finally, multiply your divisor by your quotient and add the remainder to confirm you get the dividend back.

Polynomial division practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What is polynomial division?

It splits one polynomial by another into a quotient and a remainder, just like dividing numbers. For example, (x² − 2x − 4) ÷ (x − 3) gives the quotient x + 1 and the remainder −1.

How do you do polynomial long division?

Repeat divide, multiply, subtract and bring down. For (x² − 2x − 4) ÷ (x − 3): x² ÷ x = x, x(x − 3) = x² − 3x, subtracting leaves x − 4; then x ÷ x = 1, 1(x − 3) = x − 3, and subtracting leaves −1.

How do you do synthetic division?

Use r from the divisor x − r and the coefficients of the dividend, with zeros for missing terms. For (x³ − 1) ÷ (x − 1), r = 1 and 1, 0, 0, −1 give the quotient x² + x + 1 with remainder 0.

Can you divide polynomials on a calculator?

Only calculators with a computer algebra system can divide polynomials symbolically. This tool does it with exact fractions: entering the coefficients 2, 3, −5 and 1, −1 gives the quotient 2x + 5 and remainder 0.

What does the remainder tell you?

By the remainder theorem, dividing P(x) by x − r leaves P(r). For x² − 2x − 4 divided by x − 3, P(3) = −1, the same as the remainder. A remainder of 0 means the divisor is a factor.

What is the box method for dividing polynomials?

It is an area model: the divisor's terms label the rows, and you choose column headings so the cells add up to the dividend. For (x² + 5x + 6) ÷ (x + 2), the columns are x and 3, so the quotient is x + 3 with no remainder.

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