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Polynomial Function Given Roots Calculator

Reviewed by Calculator Editorial Team

This calculator helps you find the polynomial function when you know its roots. A polynomial is a mathematical expression consisting of variables and coefficients, involving terms of the form anxn + an-1xn-1 + ... + a0. When you know the roots of a polynomial, you can construct the polynomial function using the factored form.

Introduction

Polynomial functions are fundamental in algebra and have wide applications in various fields such as physics, engineering, and economics. A root of a polynomial is a value of x that makes the polynomial equal to zero. If you know the roots of a polynomial, you can easily construct the polynomial function.

The polynomial function given its roots can be expressed in its factored form as:

P(x) = a(x - r₁)(x - r₂)...(x - rₙ)

where:

  • P(x) is the polynomial function
  • a is the leading coefficient
  • r₁, r₂, ..., rₙ are the roots of the polynomial

This calculator allows you to input the roots of a polynomial and the leading coefficient to generate the polynomial function in its expanded form.

How to Use the Calculator

  1. Enter the roots of the polynomial in the input field. Separate multiple roots with commas.
  2. Enter the leading coefficient (a) of the polynomial.
  3. Click the "Calculate" button to generate the polynomial function.
  4. The calculator will display the polynomial function in its expanded form and a visualization of the polynomial.

Note: The calculator assumes that the roots are real numbers. Complex roots are not supported in this version.

Formula

The polynomial function given its roots can be constructed using the following formula:

P(x) = a(x - r₁)(x - r₂)...(x - rₙ)

where:

  • P(x) is the polynomial function
  • a is the leading coefficient
  • r₁, r₂, ..., rₙ are the roots of the polynomial

To expand the polynomial, you can use the distributive property of multiplication over addition to multiply the factors together.

Worked Example

Let's find the polynomial function given the roots 2, -1, and 3 with a leading coefficient of 2.

Example Calculation

Given roots: 2, -1, 3

Leading coefficient (a): 2

Polynomial function in factored form:

P(x) = 2(x - 2)(x + 1)(x - 3)

Expanding the polynomial:

First, multiply (x - 2)(x + 1):

x² - x - 2

Next, multiply the result by (x - 3):

x³ - 3x² - x + 3x² - 3x + 6 = x³ - x² - 3x + 6

Finally, multiply by the leading coefficient 2:

P(x) = 2x³ - 2x² - 6x + 12

Using the calculator, you can quickly verify this result by entering the roots and leading coefficient.

FAQ

What is a polynomial function?

A polynomial function is a mathematical expression consisting of variables and coefficients, involving terms of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀. Polynomial functions are fundamental in algebra and have wide applications in various fields.

How do I find the roots of a polynomial?

The roots of a polynomial are the values of x that make the polynomial equal to zero. You can find the roots of a polynomial by solving the equation P(x) = 0. There are various methods for finding the roots of a polynomial, such as factoring, using the quadratic formula, or numerical methods.

Can I use complex roots with this calculator?

This calculator assumes that the roots are real numbers. Complex roots are not supported in this version. If you need to work with complex roots, you may need to use a more advanced calculator or software.