Polynomial Synthetic Division Calculator
Divides a polynomial by a linear factor (x − r) using synthetic division, returning the quotient coefficients and remainder. Use it to quickly test roots or factor higher-degree polynomials without long division.
About this calculator
Synthetic division is a shorthand algorithm for dividing a polynomial P(x) = aₙxⁿ + … + a₀ by a linear binomial (x − r). Instead of writing out full algebraic long division, you work only with the coefficients. The process is: bring down the leading coefficient, multiply it by r, add to the next coefficient, and repeat. By the Remainder Theorem, the final value produced is the remainder, which also equals P(r). If the remainder is zero, r is a root of the polynomial. The quotient is a polynomial of degree n−1 whose coefficients are the intermediate results generated during the sweep. This makes synthetic division far faster than long division for degree-3 and degree-4 polynomials.
How to use
Divide P(x) = 2x³ − 3x² + x − 5 by (x − 2), so divisor_root = 2 and coefficients are [2, −3, 1, −5]. Step 1: bring down 2. Step 2: 2×2 = 4; add to −3 → 1. Step 3: 1×2 = 2; add to 1 → 3. Step 4: 3×2 = 6; add to −5 → 1 (remainder). Quotient coefficients are [2, 1, 3], representing 2x² + x + 3. Remainder = 1, confirming P(2) = 1. Enter coeff_highest = 2, coeff_second = −3, coeff_third = 1, constant_term = −5, divisor_root = 2 to verify.
Frequently asked questions
How do I use synthetic division to find the roots of a polynomial?
Synthetic division is most efficient when paired with the Rational Root Theorem. First, list the possible rational roots (factors of the constant term divided by factors of the leading coefficient). Test each candidate by running synthetic division: if the remainder is zero, that candidate is a root and the quotient is a reduced polynomial. Repeat the process on the quotient to find further roots. This systematic approach factors the polynomial completely without graphing or numerical guessing.
What is the Remainder Theorem and how does it relate to synthetic division?
The Remainder Theorem states that when a polynomial P(x) is divided by (x − r), the remainder equals P(r). Synthetic division computes this remainder as the final value in the sweep, so it also evaluates the polynomial at x = r in a single efficient pass. This makes synthetic division a rapid polynomial evaluation method — often faster than substituting r directly into a large polynomial with many terms, especially by hand.
When can I not use synthetic division for polynomial long division?
Synthetic division only works when the divisor is a monic linear polynomial of the form (x − r). If the divisor is quadratic (like x² + 1) or has a leading coefficient other than 1 (like 2x − 3), synthetic division in its standard form does not apply directly. For non-monic linear divisors such as (2x − 3), you can substitute r = 3/2 and then adjust, but for any higher-degree divisor you must use full polynomial long division or other algorithms.