Synthetic Division Calculator

An online synthetic division calculator will allow you to determine the reminder and quotient of polynomials using the synthetic division method. It also finds the zeros of the denominator and the coefficient of the numerator.

In algebra mathematics, synthetic division is the way used to manually perform the Euclidean division of polynomials. The polynomials division also can solve with the long division method. But in the synthetic division, we need short writing and calculation for the answer. So, the synthetic division is the shortest method of the polynomial.

Here we discuss both methods in detail using examples.

What is Synthetic Division?

Synthetic division is a technique used to solve the division operation on polynomials when the division is a linear factor. Its benefit is it allows one to calculate without variables during performing polynomial division, So this is an easy method compared to long division 

We can represent the division of two polynomials in the form:

p(x)/q(x) = Q(x) + R/(q(x))


  • p(x) is the dividend
  • q(x) is the linear divisor
  • Q(x) is quotient
  • R is remainder

Synthetic Division Calculator

Synthetic Division of Polynomials Definition

When we divide a polynomial p(x) by a linear factor (x – a) (which is a polynomial of degree 1), Q(x) is the quotient polynomial and R is the remainder.

p(x)/q(x) = p(x)/(x- a) = Quotient + (Remainder/(x – a))

p(x)/(x – a) = Q(x) + (R/(x – a))

The coefficients of p(x) are taken and divided by the zero of the linear factor.

We use synthetic division in the context of the evaluation of the polynomials by the remainder theorem, wherein we evaluate the value of p(x) at “a” while dividing (p(x)/(x – a)). To find if “a” is the factor of the polynomial p(x), use the synthetic division to find the remainder fast. Let us comprehend this better using the example given below