Skip to content

Synthetic Division Calculator

Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form (x − c). Enter the coefficients of your polynomial and the value of c, and the calculator shows the quotient, the remainder and every step.

  • Shows every step
  • Remainder Theorem check
  • Divisors (x − c)

Your polynomial

Dividend and divisor

Highest power first, separated by commas. Use 0 for a missing power: x³ − 3x² + 5 is 1, −3, 0, 5.

For (x − 3) enter 3; for (x + 1) enter −1. You can also type x − 3.

— quotient

    Use the calculator above for your own numbers. Below, we explain how the Synthetic Division calculator works, with worked examples and sources.

    How to use the calculator

    1. Write the polynomial in descending powers of x and list its coefficients. Use 0 for any missing power — for x³ − 3x² + 5 enter 1, −3, 0, 5.
    2. Enter c from the divisor (x − c). For (x − 3) enter 3; for (x + 1) enter −1.
    3. Select Calculate to see the quotient, remainder and the working table.

    How synthetic division works

    Example: divide 2x³ − 6x² + 2x − 1 by (x − 3).

    StepActionResult
    1Bring down the first coefficient2
    22 × 3 = 6; add to −60
    30 × 3 = 0; add to 22
    42 × 3 = 6; add to −15 (remainder)

    The bottom row 2, 0, 2 gives the quotient 2x² + 0x + 2 = 2x² + 2, and the remainder is 5. So 2x³ − 6x² + 2x − 1 = (x − 3)(2x² + 2) + 5.

    Check your answer with the Remainder Theorem

    The remainder of P(x) ÷ (x − c) always equals P(c). Here P(3) = 2(27) − 6(9) + 2(3) − 1 = 5, which matches. A remainder of 0 means (x − c) is a factor of the polynomial and c is a root.

    Synthetic division vs polynomial long division

    Synthetic divisionLong division
    DivisorOnly linear (x − c)Any polynomial
    SpeedFast – coefficients onlySlower – full terms
    Best forFinding roots, factoring, evaluating P(c)Quadratic or higher divisors

    Common mistakes

    • Forgetting a 0 for a missing power (x³ + 5 needs 1, 0, 0, 5).
    • Using the wrong sign for c: (x + 2) means c = −2.
    • Dividing by ax − b without first dividing by a: for (2x − 4), use c = 2 and divide the quotient by 2.

    Sources

    1. OpenStax College Algebra 2e – 5.4 Dividing Polynomials — openstax.org/books/college-algebra-2e/pages/5-4-dividing-polynomials

    Last reviewed: . Found a problem? Email support@tuffsearch.com.

    Written & built by

    Zeeshan Nawaz

    Calculator Developer · Tuff Search

    Builds and tests Tuff Search calculators, turning official tax tables and published formulas into tools anyone can check.

    FAQ

    Frequently asked questions

    Quick answers about the Synthetic Division calculator.

    No. It only works for linear divisors (x − c). Use polynomial long division for higher-degree divisors.

    Still have a question? Email support@tuffsearch.com

    Keep exploring

    Related calculators

    More calculators people use alongside this one.