Write A Polynomial Function with Given Zeros Square Root Calculator

This calculator helps you construct a polynomial function from its given zeros. Whether you're studying algebra or need to solve a real-world problem, this tool provides a clear path to the correct polynomial equation.

Introduction

Polynomial functions are fundamental in algebra and have wide applications in science, engineering, and economics. A polynomial function with given zeros can be written using the factored form, which directly incorporates the zeros of the function.

The process involves creating factors of the form (x - r) for each real zero r, and (x² + bx + c) for complex zeros, where b and c are determined by the complex conjugate pairs. The constant term is adjusted to match the leading coefficient if specified.

How to Use the Calculator

Enter the zeros of the polynomial function in the input field, separated by commas.
Specify the leading coefficient if it's different from 1.
Click "Calculate" to generate the polynomial function.
Review the result and the step-by-step explanation.

For complex zeros, enter them as pairs (e.g., 2+3i, 2-3i). The calculator will automatically handle the complex conjugate pairs.

Formula Explained

The polynomial function with given zeros can be written in the factored form:

f(x) = a(x - r₁)(x - r₂)...(x - rₙ)

Where:

a is the leading coefficient (default is 1 if not specified)
r₁, r₂, ..., rₙ are the zeros of the polynomial

For complex zeros, the factors are grouped as (x² + bx + c), where b and c are determined by the sum and product of the complex conjugate pairs.

Worked Examples

Example 1: Simple Polynomial

Zeros: 2, -3, 5

Polynomial: f(x) = (x - 2)(x + 3)(x - 5)

Expanded form: f(x) = x³ - 2x² - 16x + 30

Example 2: Polynomial with Complex Zeros

Zeros: 1, 2+3i, 2-3i

Polynomial: f(x) = (x - 1)(x² - 4x + 13)

Expanded form: f(x) = x³ - 5x² + 17x - 13

Frequently Asked Questions

What is the difference between the factored and expanded forms of a polynomial?

The factored form shows the polynomial as a product of its factors, which directly reveals its zeros. The expanded form is a sum of terms with decreasing powers of x, which is useful for evaluating the polynomial at specific points.

How do I handle repeated zeros in the polynomial?

For repeated zeros, you include the factor (x - r) as many times as the zero appears. For example, if a zero occurs twice, you would have (x - r)² in the factored form.

Can I use this calculator for polynomials with non-integer zeros?

Yes, the calculator accepts any real or complex zeros. Simply enter the zeros as decimals or in the form a+bi for complex numbers.