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Polynomial with Given Complex Roots Calculator

Reviewed by Calculator Editorial Team

This calculator helps you find the polynomial equation given its complex roots. It's useful in algebra, engineering, and signal processing where complex roots are common.

Introduction

When you have the roots of a polynomial, you can construct the polynomial itself. This is particularly useful when dealing with complex roots, which often appear in advanced mathematical problems.

The polynomial with given roots can be found using the following relationship: if a polynomial has roots r₁, r₂, ..., rₙ, then the polynomial can be written as:

P(x) = (x - r₁)(x - r₂)...(x - rₙ)

This calculator implements this formula to generate the polynomial from your input roots.

How to Use the Calculator

  1. Enter the complex roots of your polynomial in the input fields. You can add multiple roots by clicking the "Add Root" button.
  2. Click the "Calculate" button to generate the polynomial equation.
  3. View the result in the result panel below the calculator.
  4. Use the "Reset" button to clear all inputs and start over.

Note: Complex roots should be entered in the form a + bi, where a and b are real numbers.

Formula

The polynomial with given roots r₁, r₂, ..., rₙ is constructed as:

P(x) = (x - r₁)(x - r₂)...(x - rₙ)

For complex roots, the formula remains the same, but the roots are complex numbers in the form a + bi.

Worked Example

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

  1. First root: 2
  2. Second root: -3i

The polynomial is:

P(x) = (x - 2)(x - (-3i)) = (x - 2)(x + 3i)

Expanding this gives:

P(x) = x² + 3i x - 2x - 6i = x² + (3i - 2)x - 6i

This is the polynomial with the given roots.

FAQ

What if I have more than two roots?

You can enter as many roots as needed using the "Add Root" button in the calculator. The calculator will construct the polynomial by multiplying all the factors (x - rᵢ).

Can I use this calculator for real roots?

Yes, the calculator works for both real and complex roots. Simply enter the real roots as numbers without the imaginary part.

How do I enter complex roots?

Enter complex roots in the form a + bi, where a is the real part and b is the imaginary part. For example, 2 + 3i would be entered as "2 + 3i".