Polynomial with Given Roots Calculator

A polynomial with given roots is a polynomial equation that has specific values (roots) where it equals zero. This calculator helps you find the polynomial equation when you know its roots.

What is a Polynomial with Given Roots?

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A polynomial with given roots is one where we know the values of x that make the polynomial equal to zero.

For example, if we know that x = 2 and x = 3 are roots of a polynomial, we can construct the polynomial as (x - 2)(x - 3). Expanding this gives us the polynomial x² - 5x + 6.

Note: The degree of the polynomial will be equal to the number of roots you provide. For example, two roots will give you a quadratic polynomial.

How to Use the Calculator

Using the polynomial with given roots calculator is simple:

Enter the roots of the polynomial, separated by commas. For example, "2, 3" for roots at x=2 and x=3.
Click the "Calculate" button to generate the polynomial equation.
Review the result, which will show the polynomial in both factored and expanded forms.
Use the optional chart to visualize the polynomial.

The calculator will handle up to 5 roots for practical purposes, but you can enter more if needed.

Formula and Calculation

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

If the roots are r₁, r₂, ..., rₙ, then the polynomial is:

(x - r₁)(x - r₂)...(x - rₙ)

This factored form can be expanded to standard polynomial form using the distributive property of multiplication.

For example, with roots at x=2 and x=3:

(x - 2)(x - 3) = x² - 5x + 6

Worked Example

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

Start with the factored form: (x - 1)(x - 2)(x - 3)
Multiply the first two factors: (x - 1)(x - 2) = x² - 3x + 2
Multiply the result by the third factor: (x² - 3x + 2)(x - 3) = x³ - 6x² + 11x - 6

The final polynomial is x³ - 6x² + 11x - 6.

Frequently Asked Questions

What is the difference between a polynomial and a polynomial with given roots?: A polynomial is any algebraic expression with variables and coefficients. A polynomial with given roots is a specific polynomial where we know the values of x that make it equal to zero.
Can I use negative roots with this calculator?: Yes, the calculator accepts negative roots. For example, entering "-2, 3" will give you a polynomial with roots at x=-2 and x=3.
What if I enter a repeated root?: Repeated roots are handled automatically. For example, entering "2, 2" will result in a polynomial with a double root at x=2.
Is there a limit to the number of roots I can enter?: The calculator can handle up to 5 roots, but you can enter more if needed. Very high-degree polynomials may be difficult to work with in practice.
Can I use decimal roots with this calculator?: Yes, the calculator accepts decimal roots. For example, entering "1.5, 2.5" will give you a polynomial with roots at x=1.5 and x=2.5.