Synthetic Division Calculator
Perform synthetic division of polynomials step by step with this free online calculator. Enter coefficients and a root to get quotient, remainder, and each intermediate calculation.
About This Calculator
This free Synthetic Division Calculator performs polynomial division by a linear factor x - r using the streamlined synthetic method. Instead of writing the full polynomial with variables, it works with only the coefficients arranged in a row. Enter the coefficients of your polynomial from the highest degree term down to the constant, plus the root value r, to get the quotient polynomial, remainder, and every intermediate calculation step.
The algorithm works as follows: bring down the leading coefficient, then repeatedly multiply the current result by r and add the next coefficient. The final numbers give the quotient coefficients (one degree lower than the original polynomial) and the last number is the remainder. If the remainder is zero, x - r is a factor of the polynomial and r is a root.
Synthetic division is faster than long division and is commonly used to test rational roots, factor polynomials, and evaluate polynomials using the Remainder Theorem. It is a cornerstone technique in Algebra II and Precalculus courses worldwide.
Regional Notes
India (IN): Synthetic division is introduced in Class 9 polynomials chapter under CBSE. It is extensively used in JEE preparation for quickly dividing polynomials and testing roots during factorization of cubic and quartic equations, often combined with the Rational Root Theorem and factor theorem.
United States (US): Taught in Algebra II and Precalculus courses. Students use synthetic division alongside the Rational Root Theorem to find all real zeros of polynomial functions, a key skill for AP Calculus AB/BC. It is also covered in college algebra and precalculus courses.
United Kingdom (UK): Covered at A-Level Mathematics as part of the algebra and functions module. It is used alongside the factor theorem to factorise polynomials and solve cubic equations in Core Pure Mathematics. Students applying to STEM programs frequently encounter synthetic division in university admissions tests.
Frequently Asked Questions
What is synthetic division?
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - r. It uses only the coefficients of the polynomial and performs fewer calculations than long division. It is especially useful for testing potential roots found via the Rational Root Theorem.
How does synthetic division differ from long division?
Synthetic division works only when dividing by a linear divisor (x - r), while long division works for any divisor. Synthetic division is faster because it uses only the coefficients and skips writing the variable terms. The result gives the quotient coefficients and the remainder.
When is synthetic division used?
Synthetic division is used to quickly divide polynomials by linear factors, test potential rational zeros (if the remainder is 0, the tested value is a root), evaluate polynomials at a given point via the Remainder Theorem, and factor polynomials by finding one root at a time.
What is the remainder in synthetic division?
The last number in the synthetic division result is the remainder. By the Remainder Theorem, this remainder equals the value of the polynomial evaluated at x = r (the root). If the remainder is 0, then x - r is a factor of the polynomial and r is a root.
What degree of polynomials does synthetic division work for?
Synthetic division works for polynomials of any degree as long as the divisor is linear (x - r). For quadratic or higher-degree divisors, polynomial long division is required instead.
What is the Remainder Theorem?
The Remainder Theorem states that when a polynomial f(x) is divided by x - r, the remainder equals f(r). This means synthetic division can evaluate a polynomial at a given point without substitution. For example, dividing x^3 - 2x + 4 by x - 3 gives remainder 25, so f(3) = 25.
Can synthetic division be used for quadratic or higher-degree divisors?
No, synthetic division only works for linear divisors of the form x - r. For quadratic (x^2 + bx + c) or higher-degree divisors, you must use polynomial long division instead. There is an extended synthetic division method for quadratic divisors, but it is more complex and rarely taught at the high school level.
What does a remainder of zero mean in synthetic division?
A remainder of zero means the divisor x - r divides the polynomial exactly, so r is a root (zero) of the polynomial and x - r is a factor. For example, using synthetic division to divide x^3 - 6x^2 + 11x - 6 by x - 1 gives remainder 0, confirming that 1 is a root and x - 1 is a factor of the cubic.