Polynomial Long Division Calculator

Polynomial calculator: divide polynomials; get quotient + remainder with steps.

Output: polynomial quotient and remainder.

The Polynomial Long Division Calculator divides one polynomial by another and returns the quotient and remainder with full work. It arranges both polynomials in descending order of degree, inserts zero coefficients for any missing powers, then repeats the divide–multiply–subtract–bring-down loop on whole terms. The result rewrites a rational function as a quotient polynomial plus a remainder over the divisor. Enter a dividend polynomial and a divisor polynomial to see each leading-term division, each product, each subtraction, and the final quotient and remainder in standard form.

How to use the Polynomial Long Division Calculator

To divide polynomials, follow these 4 steps:

Enter the dividend polynomial, such as 2x³ − 3x² + 4x − 5, in the first field.
Enter the divisor polynomial, such as x − 2, in the second field.
Click Calculate to divide term by term in descending order.
Read the quotient polynomial, the remainder, and each step shown.

Include zero coefficients for missing degrees, such as x³ + 0x² + 0x − 8, so the calculator keeps each power aligned in its own column.

Polynomial division and rational functions

A polynomial is a sum of terms, each a coefficient times a power of the variable. Dividing polynomials is how you simplify a rational function — a ratio of two polynomials — into a proper form. The degree of a polynomial acts like the size of a number: division proceeds from the highest degree downward. Division stops when the remainder has a lower degree than the divisor, mirroring how a numeric remainder stays below the divisor. The Polynomial Long Division Calculator manages the degrees, the placeholders, and the sign changes that make this reliable.

Division shares a total into equal groups. Long division does this digit by digit.

Divisor (32) — the number you divide by. Place it to the left of the bracket.

How polynomial long division works

The calculator divides polynomials through five internal actions:

Writes both polynomials in descending degree and fills missing powers with zero coefficients.
Divides the leading term of the dividend by the leading term of the divisor to get a quotient term.
Multiplies the whole divisor by that quotient term.
Subtracts the product, which eliminates the current leading term.
Brings down the next term and repeats until the remainder's degree is below the divisor's.

The quotient is the sum of the term-by-term results, and the leftover polynomial of lower degree is the remainder.

Writes both polynomials in descending degree and fills missing powers with zero coefficients.

Polynomial division formula

The Polynomial Long Division Calculator uses the division algorithm for polynomials: P(x) = D(x)·Q(x) + R(x), with deg(R) < deg(D). Equivalently, the rational function is P(x)/D(x) = Q(x) + R(x)/D(x). For (x² − 5x + 6) ÷ (x − 2) = x − 3, the identity confirms (x − 2)(x − 3) = x² − 5x + 6.

Dividend

Divisor

Quotient

Remainder

100 = 7 × 14 + 2 → Identity holds

Polynomial example problems

These examples divide polynomials with and without remainders.

Example 1 — Exact division: (x² − 5x + 6) ÷ (x − 2)

Divide x² ÷ x = x; multiply x(x − 2) = x² − 2x; subtract to leave −3x + 6.
Divide −3x ÷ x = −3; multiply −3(x − 2) = −3x + 6; subtract to leave 0.
So (x² − 5x + 6) ÷ (x − 2) = x − 3.

Example 2 — With a remainder: (2x³ − 3x² + 0x − 4) ÷ (x − 1)

Divide 2x³ ÷ x = 2x²; subtract 2x²(x − 1) to leave −x² + 0x − 4.
Divide −x² ÷ x = −x; subtract −x(x − 1) to leave −x − 4; divide −x ÷ x = −1; subtract to leave −5.
So the quotient is 2x² − x − 1 with remainder −5.

Example 3 — Missing powers: (x³ − 8) ÷ (x − 2)

Write x³ + 0x² + 0x − 8; divide x³ ÷ x = x²; subtract to leave 2x² + 0x − 8.
Divide 2x² ÷ x = 2x; subtract to leave 4x − 8; divide 4x ÷ x = 4; subtract to leave 0.
So (x³ − 8) ÷ (x − 2) = x² + 2x + 4.

Divide x² ÷ x = x; multiply x(x − 2) = x² − 2x; subtract to leave −3x + 6.
Divide −3x ÷ x = −3; multiply −3(x − 2) = −3x + 6; subtract to leave 0.
So (x² − 5x + 6) ÷ (x − 2) = x − 3.

Worked polynomial problems

How do you divide 3x³ + 2x² − x + 5 by x + 2?

(3x³ + 2x² − x + 5) ÷ (x + 2) = 3x² − 4x + 7 with remainder −9. Divide 3x³ ÷ x = 3x², subtract to leave −4x² − x; divide −4x² ÷ x = −4x, subtract to leave 7x + 5; divide 7x ÷ x = 7, subtract to leave −9. Check: (x + 2)(3x² − 4x + 7) − 9 = 3x³ + 2x² − x + 5.

x^3+2x^2-5x-6x-2

How does polynomial division simplify a rational function?

It rewrites P(x)/D(x) as a polynomial plus a proper fraction. For (x² + 1)/(x − 1), long division gives x + 1 with remainder 2, so (x² + 1)/(x − 1) = x + 1 + 2/(x − 1). The remainder over the divisor is the proper part, which is essential for graphing asymptotes and integrating rational functions.

x^2+5x+6x+2

Common polynomial division mistakes

Polynomial division produces 5 frequent errors:

Skipping a zero coefficient for a missing degree, which misaligns terms.
Sign errors when subtracting the product, especially with a negative divisor constant.
Multiplying the quotient term by only the leading term of the divisor.
Writing the quotient or dividend out of descending-degree order.
Stopping before the remainder's degree falls below the divisor's degree.

The Polynomial Long Division Calculator inserts placeholders, orders terms by degree, and tracks every sign so each subtraction is exact.

Skipping a zero coefficient for a missing degree, which misaligns terms.

Sign errors when subtracting the product, especially with a negative divisor constant.

Multiplying the quotient term by only the leading term of the divisor.

Writing the quotient or dividend out of descending-degree order.

Stopping before the remainder's degree falls below the divisor's degree.

Frequently Asked Questions

How do you do polynomial long division?

Divide the leading term of the dividend by the leading term of the divisor, multiply the whole divisor by that term, subtract, then bring down the next term and repeat until the remainder's degree is below the divisor's.

Why insert zero coefficients for missing terms?

A placeholder like x³ + 0x² + 0x − 8 gives each degree its own column so the subtraction stays aligned. Leaving gaps causes terms to shift and produces wrong results.

When does polynomial division stop?

It stops when the remainder polynomial has a lower degree than the divisor. That leftover of lower degree is the remainder.

How does polynomial division help with rational functions?

It rewrites an improper rational function as a quotient polynomial plus a proper remainder fraction, which reveals slant asymptotes and makes integration easier: P/D = Q + R/D.

What is the difference between long and synthetic division?

Long division works for any divisor and shows every term step. Synthetic division is a shortcut that only works when the divisor is linear, such as x − c.

How do I check a polynomial division answer?

Multiply the divisor by the quotient and add the remainder; the result must equal the dividend: P(x) = D(x)·Q(x) + R(x).

Can the remainder be a number instead of a polynomial?

Yes. When the divisor is linear, the remainder is a constant, as in (x² + 1) ÷ (x − 1) = x + 1 with remainder 2.

The polynomial division algorithm

The Polynomial Long Division Calculator satisfies P(x) = D(x)·Q(x) + R(x) with deg(R) < deg(D), the polynomial form of the division identity. For (2x³ − 3x² − 4) ÷ (x − 1) = 2x² − x − 1 R −5, the identity confirms (x − 1)(2x² − x − 1) − 5 = 2x³ − 3x² − 4.