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Polynomial Roots Calculator Wolfram

Reviewed by Calculator Editorial Team

A polynomial root is a solution to the equation P(x) = 0, where P(x) is a polynomial function. This calculator uses Wolfram's advanced algorithms to find exact and approximate roots of polynomials up to degree 10.

What is a Polynomial Root?

In mathematics, a polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. A root of a polynomial is a value of the variable that makes the polynomial equal to zero.

For a polynomial P(x) = anxn + an-1xn-1 + ... + a0, the roots are the solutions to P(x) = 0.

Polynomial roots can be real or complex numbers. For example, the quadratic equation x² - 5x + 6 = 0 has roots at x = 2 and x = 3.

How to Use This Calculator

  1. Enter the coefficients of your polynomial in the input fields. For example, for x³ - 6x² + 11x - 6, enter 1 for x³, -6 for x², 11 for x, and -6 for the constant term.
  2. Select the degree of your polynomial (up to 10).
  3. Click "Calculate Roots" to find the solutions.
  4. View the results, which include both exact and approximate roots when available.

This calculator uses Wolfram's advanced symbolic computation engine to find exact roots when possible and numerical approximations otherwise.

Formula and Calculation

The roots of a polynomial can be found using various methods depending on the polynomial's degree. For low-degree polynomials, exact solutions can be found using algebraic methods. For higher-degree polynomials, numerical methods are typically used.

For a quadratic equation ax² + bx + c = 0, the roots are given by:

x = [-b ± √(b² - 4ac)] / (2a)

For higher-degree polynomials, Wolfram's algorithms use a combination of symbolic and numerical methods to find all roots.

Worked Examples

Example 1: Quadratic Polynomial

Find the roots of x² - 5x + 6 = 0.

  1. Enter coefficients: 1 (x²), -5 (x), 6 (constant)
  2. Select degree: 2
  3. Click "Calculate Roots"
  4. Results: x = 2 and x = 3

Example 2: Cubic Polynomial

Find the roots of x³ - 6x² + 11x - 6 = 0.

  1. Enter coefficients: 1 (x³), -6 (x²), 11 (x), -6 (constant)
  2. Select degree: 3
  3. Click "Calculate Roots"
  4. Results: x = 1, x = 2, x = 3

FAQ

What is the maximum degree polynomial this calculator can solve?
This calculator can solve polynomials up to degree 10.
Does this calculator find exact roots or just approximations?
The calculator first attempts to find exact roots using symbolic methods. If exact roots cannot be found, it provides numerical approximations.
Can I use this calculator for complex polynomials?
Yes, the calculator can find roots for polynomials with complex coefficients and complex roots.
How accurate are the numerical approximations?
The numerical approximations are accurate to at least 15 decimal places.
Is there a limit to how many roots I can find?
There is no limit to the number of roots you can find, but the calculator may take longer to compute for polynomials with many roots.