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Polynomial Root Calculator Symbolab

Reviewed by Calculator Editorial Team

Finding the roots of a polynomial equation is a fundamental problem in algebra with applications in engineering, physics, and computer science. This calculator uses Symbolab's advanced symbolic computation engine to find exact and approximate roots of polynomials with coefficients of any degree.

What is a Polynomial Root?

A polynomial root (or zero) is a solution to the equation P(x) = 0, where P(x) is a polynomial function. For example, in the equation x² - 5x + 6 = 0, the roots are x = 2 and x = 3 because these values satisfy the equation.

Polynomial roots can be real or complex numbers. The Fundamental Theorem of Algebra states that an nth-degree polynomial has exactly n roots in the complex number system, counting multiplicities.

Polynomial Equation

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0

Where aₙ, aₙ₋₁, ..., a₀ are coefficients and n is the degree of the polynomial.

How to Find Polynomial Roots

There are several methods to find polynomial roots:

  1. Factoring: Express the polynomial as a product of simpler polynomials.
  2. Quadratic Formula: For second-degree polynomials (quadratics).
  3. Numerical Methods: Approximate roots using iterative techniques like Newton's method.
  4. Symbolic Computation: Use computer algebra systems like Symbolab to find exact solutions.

Note

For polynomials of degree 5 or higher, exact solutions are not always expressible in terms of radicals. In such cases, numerical methods or symbolic computation are preferred.

Using Symbolab for Polynomial Roots

Symbolab is a powerful online calculator that can find exact and approximate roots of polynomials. It uses advanced symbolic computation techniques to provide precise results.

To use Symbolab for finding polynomial roots:

  1. Enter the polynomial equation in the input field.
  2. Select the method (exact or approximate).
  3. Click "Calculate" to get the roots.

Symbolab can handle polynomials with integer, fractional, and irrational coefficients, as well as complex roots.

Example Calculation

Let's find the roots of the polynomial x³ - 6x² + 11x - 6 = 0.

  1. Factor the polynomial: (x - 1)(x - 2)(x - 3) = 0
  2. Set each factor equal to zero: x - 1 = 0, x - 2 = 0, x - 3 = 0
  3. Solve for x: x = 1, x = 2, x = 3

The roots of the polynomial are x = 1, x = 2, and x = 3.

FAQ

What is the difference between exact and approximate roots?
Exact roots are precise solutions that can be expressed in terms of radicals or other exact forms. Approximate roots are numerical estimates that may be more practical for certain applications.
Can Symbolab find roots of polynomials with complex coefficients?
Yes, Symbolab can handle polynomials with complex coefficients and find complex roots.
What if my polynomial has a high degree?
For high-degree polynomials, Symbolab will provide approximate roots or use symbolic computation to find exact forms when possible.
How accurate are the roots calculated by Symbolab?
Symbolab uses advanced algorithms to ensure high accuracy in its root calculations. For most practical purposes, the results are precise.
Can I use Symbolab for educational purposes?
Yes, Symbolab is an excellent tool for learning and teaching polynomial root calculations.