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Polynomial Long Division Calculator

Divide a polynomial by a linear divisor (x - r) using synthetic division to find the quotient and remainder.

Result

Quotient
1x^2 + -5x^1 + 6x^0
Remainder
0

Uses synthetic division, a shortcut for dividing a polynomial by a linear factor (x - r). By the Remainder Theorem, the remainder equals the value of the original polynomial evaluated at x = r, and a remainder of zero means (x - r) is a factor of the polynomial.

About the Polynomial Long Division Calculator

The Polynomial Long Division Calculator divides a polynomial by a linear factor of the form (x - r) using synthetic division, a streamlined alternative to full long division. It is built for factoring cubic and higher-degree polynomials, testing candidate rational roots, and confirming whether a given binomial divides evenly into a larger expression.

How It Works

You enter the polynomial's coefficients from highest degree to lowest, separated by commas, along with the value r from the divisor (x - r). The calculator runs synthetic division: it carries the leading coefficient down, then repeatedly multiplies the running value by r and adds the next coefficient, producing a row of quotient coefficients with one value left over as the remainder.

For coefficients a0 through an, highest to lowest degree, and divisor (x - r): b0 = a0, and each subsequent bi = ai + b(i-1) times r. The quotient's coefficients are b0 through b(n-1), and the remainder is bn. By the Remainder Theorem, that remainder always equals the original polynomial evaluated at x = r.

Formula & Methodology

For coefficients 1, -6, 11, -6, representing x cubed minus 6x squared plus 11x minus 6, divided by (x - 1): start with b0 = 1. Next: -6 plus (1 times 1) equals -5. Next: 11 plus (-5 times 1) equals 6. Next: -6 plus (6 times 1) equals 0. The quotient coefficients are 1, -5, 6, giving x squared minus 5x plus 6, and the remainder is 0, confirming (x - 1) divides evenly and is a factor of the original cubic.

Examples

Exact Division Revealing a Factor

Dividing x cubed minus 6x squared plus 11x minus 6 (coefficients 1, -6, 11, -6) by (x - 1) gives a quotient of x squared minus 5x plus 6 with a remainder of 0, confirming that (x - 1) is a factor of the cubic.

Division With a Nonzero Remainder

Dividing 2x cubed minus 3x squared plus 5 (coefficients 2, -3, 0, 5) by (x - 2) gives a quotient of 2x squared plus x plus 2 with a remainder of 9, meaning (x - 2) is not a factor.

Advantages

  • Automates the row-by-row bookkeeping of synthetic division, which is easy to make an arithmetic slip in when done by hand across several terms.
  • Immediately signals whether a given r is a root of the polynomial, since a remainder of 0 confirms it without a separate evaluation step.
  • Produces a correctly formatted quotient polynomial ready for further factoring, rather than just a numeric answer.

Common Mistakes

  • Entering the divisor's sign incorrectly: dividing by (x + 2) requires entering r = -2, not r = 2, since the method is defined for (x - r).
  • Omitting zero coefficients for missing-degree terms, which shifts every subsequent coefficient into the wrong place in the calculation.
  • Trying to use this method for divisors that are not linear binomials in x, such as dividing by a quadratic, which synthetic division does not support.

Edge Cases to Watch For

  • At least 2 coefficients are required, since synthetic division needs at minimum a linear polynomial to divide.
  • This method only works for linear divisors of the exact form (x - r); it cannot divide by higher-degree divisors or by binomials with a leading coefficient other than 1, such as (2x - 1).
  • A remainder of exactly 0 means (x - r) is a factor of the polynomial and r is therefore a root; any nonzero remainder means it is not.
  • Missing-degree terms in the original polynomial need an explicit 0 placeholder in the coefficient list, since position in the list determines each term's degree.

Common Use Cases

  • Algebra and precalculus students factoring cubic or higher-degree polynomials by testing candidate roots.
  • Anyone applying the Rational Root Theorem, using a zero remainder to confirm which candidates are actual roots.
  • Teachers and tutors quickly checking long division or synthetic division work submitted by students.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

How does the remainder relate to evaluating the polynomial directly?

The Remainder Theorem states that dividing a polynomial P(x) by (x - r) always produces a remainder equal to P(r), so this calculator's remainder output is exactly the same value you'd get by substituting r directly into the original polynomial - synthetic division is simply a faster mechanical process for finding both the quotient and that value at once.

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Conclusion

The Polynomial Long Division Calculator performs synthetic division to divide a polynomial by a linear factor, returning both the quotient and the remainder in one step. Its remainder result doubles as a fast root-testing tool through the Remainder Theorem's direct link between division and evaluation.

References