Synthetic Division Calculator
Divide polynomials using synthetic division and find quotient and remainder.
Divide polynomial by (x − r) using synthetic division
Enter all coefficients including zeros
What Is the Synthetic Division Calculator?
The Synthetic Division Calculator performs polynomial long division by a linear binomial (x − c) using the efficient synthetic division algorithm. Enter the polynomial coefficients and the root c to get the quotient polynomial coefficients and the remainder, with a full step-by-step layout.
Formula
How to Use
Enter the coefficients of the dividend polynomial (from highest to lowest degree, including zeros for missing terms). Enter the value c for the divisor (x − c). The calculator displays the synthetic division tableau with each step and outputs the quotient coefficients and remainder.
Example Calculation
Divide x³ − 6x² + 11x − 6 by (x − 2): Coefficients: [1, −6, 11, −6], c=2. Row: 1 | −4 | 3 | 0. Quotient: x² − 4x + 3, Remainder: 0. Factor: (x−2)(x²−4x+3) = (x−2)(x−1)(x−3).
Understanding Synthetic Division
Synthetic division is a compact, efficient algorithm for dividing a polynomial by a first-degree binomial (x − c). It is significantly faster than polynomial long division because it works directly with the coefficients, avoiding the repeated rewriting of variable terms. The result gives both the quotient polynomial (reduced degree) and the remainder.
The algorithm follows a simple pattern: write the coefficients, bring down the first one, multiply by c and add to the next coefficient, repeat. The final number in the row is the remainder. By the Remainder Theorem, this is also the value of the polynomial at x = c, providing a fast evaluation method.
Synthetic division is used to verify polynomial roots, factor polynomials step by step using the Rational Root Theorem candidates, deflate polynomials after finding a root, and evaluate polynomials at specific values (Horner's method). It is an essential technique for algebra and pre-calculus students working with higher-degree polynomials.
Frequently Asked Questions
When can I use synthetic division?
Synthetic division only works when dividing by a linear factor of the form (x − c), where c is a constant. For divisors of higher degree, use polynomial long division.
What does a zero remainder mean?
A zero remainder means c is a root of the polynomial, and (x − c) is a factor. This confirms the relationship P(c) = 0 (Factor Theorem).
What if the divisor is (ax − b)?
If the divisor is (ax − b), first find c = b/a and divide by (x − b/a). The quotient must then be adjusted: divide by a to get the true quotient.
How do I handle missing terms (e.g., x³ + 2)?
Insert zero coefficients for any missing degree terms. For x³ + 2, use coefficients [1, 0, 0, 2].
Is this calculator free?
Yes, completely free with no registration needed.
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