Roots of Polynomials Calculator

Find the roots of any polynomial equation with our accurate calculator. Whether you're solving quadratic, cubic, or higher-degree polynomials, this tool provides precise solutions and explains the underlying mathematics.

How to Use This Calculator

Using our roots of polynomials calculator is simple:

Enter the coefficients of your polynomial in the input fields. For example, for the polynomial 2x³ - 5x² + 3x - 7, you would enter 2 for x³, -5 for x², 3 for x, and -7 for the constant term.
Select the degree of your polynomial from the dropdown menu.
Click the "Calculate" button to find the roots.
Review the results, which will display all real and complex roots of your polynomial.

The calculator will display the roots in both decimal and exact form where possible. For complex roots, it will show both the real and imaginary parts.

How Polynomial Roots Are Calculated

The roots of a polynomial are the values of x that satisfy the equation P(x) = 0. Finding these roots is a fundamental problem in algebra with applications in many fields.

Methods for Finding Roots

There are several methods for finding polynomial roots, each with different advantages depending on the polynomial's degree and nature of the roots:

Quadratic Formula: For second-degree polynomials (ax² + bx + c), the roots can be found using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a).
Cubic Formula: For third-degree polynomials, there is a more complex formula involving radicals and trigonometric functions.
Numerical Methods: For higher-degree polynomials or when exact solutions are difficult to find, numerical methods like Newton's method or the bisection method can approximate the roots.

For a general polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, the roots are the solutions to P(x) = 0.

Our calculator uses a combination of exact methods for low-degree polynomials and numerical approximation for higher-degree cases to provide accurate results.

Worked Examples

Example 1: Quadratic Polynomial

Find the roots of 2x² - 5x + 3 = 0.

Identify the coefficients: a = 2, b = -5, c = 3.
Calculate the discriminant: D = b² - 4ac = (-5)² - 4(2)(3) = 25 - 24 = 1.
Apply the quadratic formula: x = [5 ± √1] / 4.
Find the roots: x₁ = (5 + 1)/4 = 1.5, x₂ = (5 - 1)/4 = 1.

The roots are x = 1.5 and x = 1.

Example 2: Cubic Polynomial

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

Identify the coefficients: a = 1, b = -6, c = 11, d = -6.
Use the cubic formula or factor the polynomial: (x - 1)(x - 2)(x - 3) = 0.
Find the roots: x = 1, x = 2, x = 3.

The roots are x = 1, x = 2, and x = 3.

Frequently Asked Questions

What is a polynomial root?

A polynomial root is a value of x that makes the polynomial equal to zero. For example, in the equation x² - 4 = 0, the roots are x = 2 and x = -2.

How many roots can a polynomial have?

A polynomial of degree n can have up to n roots, counting multiplicities. For example, a quadratic polynomial can have 2 roots, and a cubic polynomial can have 3 roots.

What if my polynomial has complex roots?

Our calculator will display complex roots in the form a + bi, where a is the real part and b is the imaginary part. Complex roots come in conjugate pairs for polynomials with real coefficients.

Can this calculator solve polynomials with non-integer coefficients?

Yes, our calculator can handle polynomials with any real coefficients, including decimals and fractions. Simply enter the coefficients as you would for any other polynomial.