Polynomials Roots Calculator

This polynomials roots calculator finds all real and complex roots of any polynomial equation. Enter your polynomial coefficients and get exact solutions using numerical methods. The calculator handles quadratic, cubic, and higher-degree polynomials with clear results and visualizations.

What is a polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form is:

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

Where:

aₙ, aₙ₋₁, ..., a₀ are coefficients
x is the variable
n is the degree of the polynomial

Polynomials can have real or complex roots, which are the values of x that satisfy P(x) = 0.

How to find polynomial roots

Finding roots of polynomials is a fundamental problem in algebra with applications in physics, engineering, and computer science. The methods vary depending on the polynomial's degree:

Quadratic (degree 2): Use the quadratic formula
Cubic (degree 3): Use Cardano's formula
Higher degrees: Use numerical methods like Newton-Raphson

For polynomials of degree 5 or higher, exact solutions are generally not possible, so numerical approximation methods are used.

Methods for finding roots

Quadratic Formula

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

x = [-b ± √(b² - 4ac)] / (2a)

Numerical Methods

For higher-degree polynomials, numerical methods like:

Bisection method
Newton-Raphson method
Secant method

are used to approximate roots. These methods iteratively improve the guess for the root until it reaches a desired accuracy.

Worked examples

Example 1: Quadratic Polynomial

Find the roots of x² - 5x + 6 = 0.

Using the quadratic formula:

a = 1, b = -5, c = 6 x = [5 ± √(25 - 24)] / 2 x = [5 ± 1] / 2 Roots: x = 3 and x = 2

Example 2: Cubic Polynomial

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

Using numerical approximation methods, we find the roots:

x ≈ 1, x ≈ 2, x ≈ 3

FAQ

What is the difference between real and complex roots?

Real roots are actual numbers that satisfy the equation, while complex roots have imaginary components. For example, x² + 1 = 0 has complex roots x = ±i.

Can this calculator solve all polynomials?

Yes, the calculator uses numerical methods to find roots of any polynomial, including those with complex roots.

How accurate are the results?

The calculator provides results with high precision, typically within 10⁻⁶ of the actual root value.