Write A Polynomial Equation with Roots Given Calculator

This calculator helps you construct a polynomial equation from given roots. Whether you're a student studying algebra or a professional working with polynomial functions, this tool provides a quick and accurate way to generate the equation based on the roots you provide.

Introduction

A polynomial equation is a mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. When you know the roots of a polynomial, you can write the equation in its factored form.

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. For example, if you know that a polynomial has roots at x = 2 and x = -3, you can write the polynomial as (x - 2)(x + 3). Expanding this gives you the standard form of the polynomial equation.

How to Use the Calculator

Using the calculator is straightforward. Follow these steps:

Enter the roots of your polynomial in the input field. Separate multiple roots with commas.
Click the "Calculate" button to generate the polynomial equation.
Review the result, which will be displayed in both factored and expanded forms.
If needed, you can reset the calculator to start over.

The calculator will handle both integer and decimal roots, and it will provide the equation in its simplest form.

Formula Explained

The polynomial equation with given roots can be written in its factored form as:

P(x) = (x - r₁)(x - r₂)...(x - rₙ)

Where r₁, r₂, ..., rₙ are the roots of the polynomial. To convert this to the standard form, you would expand the product of the factors.

For example, if the roots are 2 and -3, the factored form is (x - 2)(x + 3), and the expanded form is x² + x - 6.

Worked Examples

Example 1: Roots at x = 1 and x = -2

Factored form: (x - 1)(x + 2)

Expanded form: x² + x - 2

Example 2: Roots at x = 0.5 and x = -1.5

Factored form: (x - 0.5)(x + 1.5)

Expanded form: x² + x - 0.75

Example 3: Roots at x = 1, x = -1, and x = 2

Factored form: (x - 1)(x + 1)(x - 2)

Expanded form: x³ - 2x² - x + 2

Frequently Asked Questions

What is the difference between the factored and expanded forms of a polynomial?: The factored form of a polynomial shows the roots as factors, while the expanded form is a single polynomial expression. The expanded form is often easier to evaluate for specific values of x.
Can the calculator handle complex roots?: Currently, the calculator handles real roots. For complex roots, you would need to use a more advanced mathematical tool.
How do I enter multiple roots into the calculator?: Separate each root with a comma in the input field. For example, to enter roots at x = 1 and x = -2, you would type "1, -2".
What if I enter a root that is not a number?: The calculator will display an error message if you enter a non-numeric value. Please ensure all roots are valid numbers.
Can I use this calculator for higher-degree polynomials?: Yes, the calculator can handle polynomials of any degree, as long as you provide the correct roots.