Write A Polynomial Function with Given Roots Calculator

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

Introduction

A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. When given the roots of a polynomial, we can construct the polynomial using the factored form.

The Fundamental Theorem of Algebra states that every non-zero, single-variable, degree n polynomial with complex number coefficients has, counted with multiplicity, exactly n roots. This means that if we know all the roots of a polynomial, we can write the polynomial in its factored form.

How to Use the Calculator

Enter the roots of the polynomial, separated by commas. For example, if the roots are 2, -3, and 5, enter "2, -3, 5".
Select whether you want the polynomial to be monic (leading coefficient of 1) or not.
Click the "Calculate" button to generate the polynomial function.
The result will display the polynomial in both factored and expanded forms, along with a graph of the polynomial.

Note: The calculator assumes all roots are real numbers. For complex roots, you can enter them in the form "a+bi" or "a-bi".

Method: Constructing a Polynomial from Roots

To construct a polynomial from its roots, we use the factored form of the polynomial. The factored form is a product of linear factors, each corresponding to one of the roots.

If the roots are \( r_1, r_2, \ldots, r_n \), then the polynomial can be written as:

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

where \( a \) is the leading coefficient. If the polynomial is monic, \( a = 1 \).

Once the polynomial is in factored form, it can be expanded to standard form using the distributive property.

Worked Example

Let's construct a polynomial with roots at \( x = 2 \), \( x = -3 \), and \( x = 5 \).

Write the factored form of the polynomial: \( P(x) = (x - 2)(x + 3)(x - 5) \).
Expand the polynomial:

First, multiply \( (x - 2)(x + 3) \): \( x^2 + 3x - 2x - 6 = x^2 + x - 6 \).
Next, multiply the result by \( (x - 5) \): \( (x^2 + x - 6)(x - 5) = x^3 - 5x^2 + x^2 - 5x - 6x + 30 = x^3 - 4x^2 - 11x + 30 \).

The final polynomial is \( P(x) = x^3 - 4x^2 - 11x + 30 \).

Note: The expanded form is not unique. Different orderings of multiplication can lead to equivalent but differently ordered terms.

FAQ

What is a polynomial function?: A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
How do I enter complex roots?: You can enter complex roots in the form "a+bi" or "a-bi". For example, if the roots are 2, -3, and 1+2i, enter "2, -3, 1+2i".
What is a monic polynomial?: A monic polynomial is a polynomial where the leading coefficient (the coefficient of the highest power of x) is 1.
Can I use this calculator for polynomials with repeated roots?: Yes, you can enter repeated roots by listing them multiple times. For example, if the roots are 2, 2, and 5, enter "2, 2, 5".
Is the expanded form of the polynomial unique?: The expanded form is not unique. Different orderings of multiplication can lead to equivalent but differently ordered terms. However, the factored form is unique up to the order of the factors.