What do I get dividing polynomials?
Type the dividend and the divisor. The dividing polynomials calculator does the long division in exact fractions and writes out each step.
- Quotient
- 2x^2 + 5x - 7
The quotient is 2x^2 + 5x - 7 with remainder 1.
- Remainder
- 1
- Division algorithm
- 6x^3 + 11x^2 - 31x + 15 = (3x - 2) × (2x^2 + 5x - 7) + 1
- Steps
- Divide 6x^3 by 3x: 2x^2. Multiply: 2x^2 × (3x - 2) = 6x^3 - 4x^2. Subtract: 15x^2 - 31x + 15; Divide 15x^2 by 3x: 5x. Multiply: 5x × (3x - 2) = 15x^2 - 10x. Subtract: -21x + 15; Divide -21x by 3x: -7. Multiply: -7 × (3x - 2) = -21x + 14. Subtract: 1
Quotient: 2x^2 + 5x - 7. The quotient is 2x^2 + 5x - 7 with remainder 1.
How is the division done?
How to calculate
Divides one polynomial by another by long division in exact fractions, showing every step, the quotient and the remainder, and synthetic division when the divisor is x − c.
Example with the default inputs (Dividend 6x^3 + 11x^2 - 31x + 15, Divisor 3x - 2): The quotient is 2x^2 + 5x - 7 with remainder 1.
Method: Long division: divide the leading terms, multiply the divisor by that term, subtract, and repeat until the remainder’s degree is below the divisor’s.
- Coefficients are exact fractions: 0.1 is 1/10.
- One letter, whole powers up to 50.
Worked examples
Each example is checked against the calculator on every build.
- Dividend 6x^3 + 11x^2 - 31x + 15, Divisor 3x - 2 gives Quotient 2x^2 + 5x - 7, Remainder 1, Division algorithm 6x^3 + 11x^2 - 31x + 15 = (3x - 2) × (2x^2 + 5x - 7) + 1.Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02): (6x³ + 11x² − 31x + 15) ÷ (3x − 2) = 2x² + 5x − 7 remainder 1
- Dividend 5x^2 + 3x - 2, Divisor x + 1 gives Quotient 5x - 2, Remainder 0, Synthetic division c = -1; bottom row: 5, -2, 0.Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02): (5x² + 3x − 2) ÷ (x + 1) = 5x − 2, remainder 0
- Dividend x^3 - 1, Divisor 2x^2 + 1 gives Quotient x/2, Remainder -x/2 - 1, Division algorithm x^3 - 1 = (2x^2 + 1) × x/2 - x/2 - 1.Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02)
How it works
Type the dividend and the divisor as polynomials in one lowercase letter other than e (the same letter in both), with whole powers from 0 to 50, for example 6x^3 + 11x^2 - 31x + 15. Brackets and products are multiplied out first. Every coefficient is read as an exact fraction (0.1 is 1/10). There is no answer when the divisor is 0, when a box is not a polynomial in one letter, or when the two boxes use different letters.
Long division. With the dividend’s degree m and the divisor’s degree k, for each power from m − k down to 0:
- Divide the current leading term (the term of that power plus k) by the divisor’s leading term.
- Multiply the divisor by the result and subtract it from what is left.
The results are the quotient’s terms; what is left at the end, with degree below k, is the remainder. If the dividend’s degree is below the divisor’s, the quotient is 0 and the remainder is the dividend.
The page shows:
- Quotient and remainder in exact fractions, with terms from the highest power down (x/2 for ½x).
- Division algorithm: dividend = (divisor) × (quotient) + remainder; the remainder part is left out when it is 0.
- Synthetic division, when the divisor is x − c (degree 1 with leading coefficient 1): c, then the bottom row, b₁ = first coefficient and each next b = coefficient + c × previous b. The last number is the remainder.
- Steps: for each quotient term, "Divide … by …", "Multiply: …" and "Subtract: …" (what is left, written in full). Steps whose leading term is 0 are skipped, and at most 25 steps are shown.
Worked examples by hand
(6x³ + 11x² − 31x + 15) ÷ (3x − 2). 6x³ ÷ 3x = 2x²; 2x²(3x − 2) = 6x³ − 4x²; subtract: 15x² − 31x + 15. 15x² ÷ 3x = 5x; 5x(3x − 2) = 15x² − 10x; subtract: −21x + 15. −21x ÷ 3x = −7; −7(3x − 2) = −21x + 14; subtract: 1. Quotient 2x² + 5x − 7, remainder 1.
(5x² + 3x − 2) ÷ (x + 1). Quotient 5x − 2, remainder 0. Synthetic division with c = −1: bring down 5; −1 × 5 + 3 = −2; −1 × (−2) − 2 = 0, so the bottom row is 5, −2, 0.
(x³ − 1) ÷ (2x² + 1). x³ ÷ 2x² = x/2; (x/2)(2x² + 1) = x³ + x/2; subtract: −x/2 − 1, which has degree 1, below 2. Quotient x/2, remainder −x/2 − 1.
Other questions people ask
How do I divide polynomials by long division?
Divide the leading term of the dividend by the leading term of the divisor, multiply the whole divisor by the result, subtract, and repeat with what is left. Stop when what is left has a lower degree than the divisor: that is the remainder.
What is the division algorithm for polynomials?
For a dividend f(x) and a divisor d(x) that is not 0, there are unique polynomials q(x) and r(x) with f(x) = d(x) q(x) + r(x), where r(x) is 0 or has a lower degree than d(x). The calculator shows this line for your answer.
What is synthetic division?
A shortcut for dividing by x − c. Write the dividend’s coefficients, bring down the first, then repeatedly multiply by c and add to the next coefficient. The last number is the remainder; the others are the quotient’s coefficients.
What does a remainder of 0 mean?
The divisor is a factor of the dividend. (5x² + 3x − 2) ÷ (x + 1) leaves 0, so 5x² + 3x − 2 = (x + 1)(5x − 2).
How does the remainder theorem help?
When you divide by x − c, the remainder equals the dividend’s value at x = c. Dividing −9x⁴ + 10x³ + 7x² − 6 by x − 1 leaves 2, and −9 + 10 + 7 − 6 = 2.
Can the coefficients be fractions or decimals?
Yes. Every coefficient is an exact fraction (0.5 is 1/2), so the quotient and remainder are exact. x³ − 1 divided by 2x² + 1 gives x/2 with remainder −x/2 − 1.