Synthetic Division Calculator
Calculate synthetic division instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Calculator
Adjust values & calculateEnter coefficients in descending order. Include 0 for missing terms.
Step-by-Step Synthetic Division
| c = 2 | 1 | -6 | 11 | -6 |
|---|---|---|---|---|
| multiply | 2 | -8 | 6 | |
| result | 1 | -4 | 3 | 0 |
Formula
Synthetic division works by repeatedly multiplying the current result by the divisor value c and adding to the next coefficient. The last number in the bottom row is the remainder, and all preceding numbers are the quotient coefficients. If the remainder is zero, then (x - c) is a factor of P(x).
Last reviewed: December 2025
Worked Examples
Example 1: Dividing a Cubic Polynomial
Example 2: Testing a Potential Root
Background & Theory
The Synthetic Division Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Synthetic Division Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
P(x) / (x - c): bring down a_n, then multiply by c and add to next coefficient
Synthetic division works by repeatedly multiplying the current result by the divisor value c and adding to the next coefficient. The last number in the bottom row is the remainder, and all preceding numbers are the quotient coefficients. If the remainder is zero, then (x - c) is a factor of P(x).
Worked Examples
Example 1: Dividing a Cubic Polynomial
Problem: Divide P(x) = x^3 - 6x^2 + 11x - 6 by (x - 2) using synthetic division.
Solution: Coefficients: 1, -6, 11, -6 and divisor c = 2\nBring down 1\n1 * 2 = 2, add to -6: -4\n-4 * 2 = -8, add to 11: 3\n3 * 2 = 6, add to -6: 0\nBottom row: 1, -4, 3, 0
Result: Quotient: x^2 - 4x + 3, Remainder: 0. Since remainder is 0, x = 2 is a root and (x - 2) is a factor.
Example 2: Testing a Potential Root
Problem: Use synthetic division to test whether x = 3 is a root of P(x) = 2x^3 - 5x^2 - 4x + 3.
Solution: Coefficients: 2, -5, -4, 3 and divisor c = 3\nBring down 2\n2 * 3 = 6, add to -5: 1\n1 * 3 = 3, add to -4: -1\n-1 * 3 = -3, add to 3: 0\nBottom row: 2, 1, -1, 0
Result: Quotient: 2x^2 + x - 1, Remainder: 0. x = 3 IS a root. Factor further: (x - 3)(2x - 1)(x + 1).
Frequently Asked Questions
What is synthetic division and when should I use it?
Synthetic division is a shorthand method for dividing a polynomial by a linear factor of the form (x - c). It dramatically simplifies the traditional long division process by working only with the coefficients of the polynomial rather than writing out full variable expressions. This method is particularly useful when you need to test potential roots of a polynomial using the Rational Root Theorem, or when factoring higher-degree polynomials step by step. Synthetic division reduces errors and saves significant time compared to polynomial long division, making it the preferred approach in most algebra courses.
How do I set up synthetic division correctly?
To set up synthetic division, first write down all coefficients of the polynomial in descending order of degree, including zeros for any missing terms. For example, x^3 + 2x - 5 has coefficients 1, 0, 2, -5 because the x^2 term is missing and needs a zero placeholder. Then write the value c from the divisor (x - c) to the left side. If you are dividing by (x + 3), use c = -3 since (x + 3) = (x - (-3)). Getting the sign correct on the divisor value is the most common source of errors in synthetic division problems.
What does the remainder in synthetic division tell me?
The remainder in synthetic division has a direct connection to the Remainder Theorem, which states that when polynomial P(x) is divided by (x - c), the remainder equals P(c). This means you can evaluate any polynomial at a specific value simply by performing synthetic division. If the remainder is zero, then c is a root of the polynomial and (x - c) is a factor. This connection between division and evaluation makes synthetic division an incredibly efficient tool for finding polynomial roots, testing potential rational roots, and completely factoring polynomials into linear factors.
Can synthetic division be used with any divisor?
Standard synthetic division only works when dividing by a linear expression of the form (x - c). You cannot directly use synthetic division to divide by quadratic or higher-degree polynomials like (x^2 + 1) or (x^2 - 3x + 2). For those cases, you must use polynomial long division instead. However, there is an extended version of synthetic division that can handle divisors of the form (ax - c) where a is not equal to 1, though it requires an additional step of dividing all quotient coefficients by a at the end. Most textbooks stick to the standard form where the leading coefficient of the divisor is one.
How does synthetic division relate to the Factor Theorem?
The Factor Theorem is a special case of the Remainder Theorem and states that (x - c) is a factor of polynomial P(x) if and only if P(c) = 0. When you perform synthetic division and get a remainder of zero, you have simultaneously proven that (x - c) is a factor AND found the quotient polynomial. This allows you to completely factor polynomials by repeatedly applying synthetic division. For instance, if you have a cubic polynomial and find one root, synthetic division gives you a quadratic quotient that you can then solve using the quadratic formula to find the remaining roots.
What are common mistakes students make with synthetic division?
The most frequent error is using the wrong sign for the divisor. When dividing by (x + 3), students often use +3 instead of the correct value -3. Another common mistake is forgetting to include a zero coefficient for missing degree terms in the polynomial. For example, dividing x^4 - 1 by (x - 1) requires coefficients 1, 0, 0, 0, -1 with three zero placeholders. Students also sometimes forget to bring down the first coefficient or make arithmetic errors when multiplying and adding. Always double-check by verifying that the quotient times the divisor plus the remainder equals the original polynomial.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy