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Synthetic Division Root Calculator

Reviewed by Calculator Editorial Team

Synthetic division is a simplified method for dividing polynomials by binomials of the form (x - c). This calculator helps you find polynomial roots using synthetic division, providing both the quotient and the remainder.

What is Synthetic Division?

Synthetic division is an efficient shortcut for polynomial long division when dividing by a binomial of the form (x - c). It's particularly useful for finding roots of polynomials, which are values of x that make the polynomial equal to zero.

The method works by systematically evaluating the polynomial at x = c, which helps determine if c is a root and what the quotient polynomial would be if you divided the original polynomial by (x - c).

For a polynomial P(x) = anxn + an-1xn-1 + ... + a1x + a0, synthetic division by (x - c) yields:

Quotient: Q(x) = anxn-1 + bn-1xn-2 + ... + b1x + b0

Remainder: R = a0 - c*b0

If the remainder R is zero, then c is a root of the polynomial P(x).

How to Use the Calculator

  1. Enter the coefficients of your polynomial in the "Coefficients" field, separated by commas (e.g., "6,11,6" for 6x² + 11x + 6)
  2. Enter the value of c (the potential root you're testing) in the "Test Root" field
  3. Click "Calculate" to perform the synthetic division
  4. Review the results showing the quotient polynomial and remainder

Note: The calculator assumes the polynomial is written in standard form with descending powers of x.

Step-by-Step Guide

Step 1: Write Down the Coefficients

List all coefficients of the polynomial in order from highest degree to lowest. For example, for 3x³ + 2x² - 5x + 1, the coefficients are [3, 2, -5, 1].

Step 2: Set Up the Synthetic Division Table

Create a table with two rows. The top row contains the coefficients, and the bottom row will hold the intermediate results.

Step 3: Bring Down the Leading Coefficient

Write the first coefficient (the leading coefficient) in the bottom row.

Step 4: Multiply and Add

For each subsequent coefficient:

  1. Multiply the value just written in the bottom row by c
  2. Add this product to the next coefficient from the top row
  3. Write the result in the bottom row

Step 5: Interpret the Results

The last number in the bottom row is the remainder. If it's zero, c is a root. The numbers in the bottom row (excluding the first) represent the coefficients of the quotient polynomial.

Example Calculation

Let's find if x = 2 is a root of P(x) = 6x³ + 11x² + 7x + 2.

Coefficients: [6, 11, 7, 2]

Test root: c = 2

Using the calculator, we get:

  • Quotient: 6x² + 23x + 51
  • Remainder: 106

Since the remainder is not zero, x = 2 is not a root of this polynomial.

FAQ

What is the difference between synthetic and polynomial long division?
Synthetic division is a shortcut method that works only when dividing by binomials of the form (x - c). Polynomial long division can handle any divisor but is more time-consuming.
How do I know if c is a root of the polynomial?
If the remainder from synthetic division is zero, then c is a root of the polynomial.
Can synthetic division be used for complex roots?
Yes, but the process is more complex. The calculator currently works with real numbers only.
What if my polynomial has a missing term?
Include zero coefficients for the missing terms. For example, x³ + 5x would be represented as [1, 0, 0, 5].