Write A Polynomial with The Given Roots Calculator

This calculator helps you construct a polynomial equation from its given roots. Whether you're a student studying algebra or a professional working with polynomial functions, this tool provides a quick and accurate way to write a polynomial with specific roots.

How to Use the Calculator

Using our polynomial roots calculator is simple. Follow these steps:

Enter the roots of the polynomial in the input field. Separate multiple roots with commas.
Select the leading coefficient if you want a specific leading coefficient for your polynomial.
Click the "Calculate" button to generate the polynomial equation.
Review the result and use it in your calculations or studies.

The calculator will display the polynomial in its standard form, such as (x - r₁)(x - r₂)...(x - rₙ).

How It Works

The calculator uses the factor theorem to construct a polynomial from its roots. The factor theorem states that if a polynomial has a root at x = r, then (x - r) is a factor of the polynomial.

Formula

If a polynomial has roots r₁, r₂, ..., rₙ, then the polynomial can be written as:

P(x) = a(x - r₁)(x - r₂)...(x - rₙ)

where 'a' is the leading coefficient (default is 1 if not specified).

The calculator expands this factored form into the standard polynomial form by multiplying the factors together.

Examples

Example 1: Simple Roots

If the roots are 2 and 3, the polynomial is:

(x - 2)(x - 3) = x² - 5x + 6

Example 2: Complex Roots

If the roots are 1, -1, and 2, the polynomial is:

(x - 1)(x + 1)(x - 2) = x³ - 2x² - x + 2

Example 3: With Leading Coefficient

If the roots are 1 and 2 with a leading coefficient of 3, the polynomial is:

3(x - 1)(x - 2) = 3x² - 9x + 6

FAQ

What is the difference between roots and coefficients?: Roots are the values of x that make the polynomial equal to zero. Coefficients are the numbers that multiply the powers of x in the polynomial equation.
Can I use negative roots?: Yes, the calculator accepts negative roots. Simply enter the negative value as a root, and the calculator will handle it correctly.
What if I have repeated roots?: The calculator will create factors for each root, including repeated roots. For example, roots at 2 and 2 will result in (x - 2)².
Is the leading coefficient optional?: Yes, the leading coefficient is optional. If you don't specify it, the calculator will use a default value of 1.