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Roots to Polynomial Calculator A 1

Reviewed by Calculator Editorial Team

This calculator helps you convert a set of roots into a polynomial equation. Understanding how roots relate to polynomial coefficients is fundamental in algebra and has applications in engineering, physics, and computer science.

What is Roots to Polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients, involving terms of the form anxn + an-1xn-1 + ... + a0. The roots of a polynomial are the values of x that satisfy the equation when set to zero.

The relationship between roots and coefficients is governed by Vieta's formulas, which state that for a polynomial with roots r₁, r₂, ..., rₙ, the sum of the roots is equal to the negative of the coefficient of xn-1 divided by the leading coefficient, and the product of the roots is equal to the constant term divided by the leading coefficient.

For a polynomial with roots r₁, r₂, ..., rₙ, the polynomial can be expressed as:

(x - r₁)(x - r₂)...(x - rₙ) = 0

How to Use This Calculator

  1. Enter the roots of your polynomial in the input field, separated by commas.
  2. Click the "Calculate" button to generate the polynomial equation.
  3. Review the result, which will show the polynomial in expanded form.
  4. Use the "Reset" button to clear the inputs and start over.

Mathematical Formula

The polynomial with roots r₁, r₂, ..., rₙ can be constructed using the following formula:

P(x) = (x - r₁)(x - r₂)...(x - rₙ)

This formula represents the polynomial in its factored form. To convert it to expanded form, you would multiply the factors together.

Example Calculation

Let's find the polynomial with roots 2 and 3.

  1. Enter the roots as "2, 3" in the calculator.
  2. Click "Calculate".
  3. The result will show the polynomial in factored form: (x - 2)(x - 3).
  4. Expanding this gives: x² - 5x + 6.

Note: The calculator shows the polynomial in factored form. For the expanded form, you would need to multiply the factors manually or use a polynomial expansion calculator.

Frequently Asked Questions

What is the difference between factored and expanded form?
Factored form shows the polynomial as a product of its factors, while expanded form shows it as a sum of terms with decreasing powers of x.
Can I use negative roots with this calculator?
Yes, you can enter negative roots by prefixing them with a minus sign (e.g., "-2, 3").
How many roots can I enter?
You can enter as many roots as you need, separated by commas. The calculator will handle them appropriately.