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Write The Simplest Polynomial Function with The Given Roots Calculator

Reviewed by Calculator Editorial Team

This calculator helps you write the simplest polynomial function with given roots. A polynomial is a mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

Introduction

When you have the roots of a polynomial, you can construct the polynomial function itself. The simplest polynomial with given roots is called the minimal polynomial, and it's constructed by multiplying linear factors corresponding to each root.

If the roots are \( r_1, r_2, \ldots, r_n \), then the polynomial is:

\( P(x) = (x - r_1)(x - r_2)\cdots(x - r_n) \)

This calculator will help you expand this product form into a standard polynomial form.

How to Use the Calculator

  1. Enter the roots of your polynomial, separated by commas (e.g., 2, -1, 3)
  2. Click "Calculate" to generate the polynomial
  3. View the result in both factored and expanded forms
  4. Use the chart to visualize the polynomial

Method

The calculator uses the following steps to construct the polynomial:

  1. Parse the input roots into an array of numbers
  2. Construct the factored form \( P(x) = (x - r_1)(x - r_2)\cdots(x - r_n) \)
  3. Expand the product to get the standard polynomial form
  4. Simplify the polynomial by combining like terms
  5. Display both forms and visualize the polynomial

Note: The calculator assumes all roots are real numbers. Complex roots would require different handling.

Example Calculation

Let's find the polynomial with roots at 1, -2, and 3.

Factored form:

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

Expanding this:

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

\( x^2 + 2x - x - 2 = x^2 + x - 2 \)

Now multiply by \( (x - 3) \):

\( (x^2 + x - 2)(x - 3) = x^3 - 3x^2 + x^2 - 3x - 2x + 6 \)

Combine like terms:

\( x^3 - 2x^2 - 5x + 6 \)

The final polynomial is \( x^3 - 2x^2 - 5x + 6 \).

FAQ

What if I have complex roots?

This calculator currently only handles real roots. For complex roots, you would need to use conjugate pairs and handle the imaginary components separately.

Can I enter repeated roots?

Yes, you can enter the same root multiple times. The calculator will account for the multiplicity in the polynomial.

What's the difference between factored and expanded form?

The factored form shows the polynomial as a product of its roots, while the expanded form shows it as a sum of terms with coefficients. The expanded form is often easier to evaluate for specific x values.

How accurate are the calculations?

The calculator uses precise arithmetic operations to ensure accurate results. However, for very large or very small numbers, floating-point precision limitations may apply.