Polynomial Calculator Given Roots

Introduction

A polynomial is a mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation. When you know the roots of a polynomial, you can determine the polynomial itself.

The Fundamental Theorem of Algebra states that every non-zero, single-variable, degree n polynomial with complex number coefficients has, counted with multiplicity, exactly n roots. This means that if you know all the roots of a polynomial, you can construct the polynomial equation.

How to Use the Calculator

Using our polynomial calculator given roots is simple:

1. Enter the roots of the polynomial in the input field, separated by commas.
2. Select the degree of the polynomial (optional).
3. Click the "Calculate" button to generate the polynomial equation.
4. Review the result and the polynomial graph.

The calculator will display the polynomial equation in its standard form, such as (x - r₁)(x - r₂)...(x - rₙ).

Formula

Given a set of roots {r₁, r₂, ..., rₙ}, the polynomial can be constructed as:

This formula represents the polynomial in its factored form, where each factor (x - rᵢ) corresponds to a root of the polynomial.

For example, if the roots are 2, -1, and 3, the polynomial would be:

Worked Example

Let's find the polynomial given the roots 1, -2, and 4.

1. Identify the roots: r₁ = 1, r₂ = -2, r₃ = 4.
2. Apply the formula: P(x) = (x - 1)(x + 2)(x - 4).
3. Expand the polynomial to standard form:
   P(x) = x³ - 3x² - 6x + 8

The polynomial equation is x³ - 3x² - 6x + 8.