1. What is a Polynomial from Zeros?

A polynomial can be constructed from its zeros (roots) using the factored form. Given the zeros r₁, r₂, ..., rₙ and a leading coefficient a, the polynomial is expressed as P(x) = a(x - r₁)(x - r₂)...(x - rₙ).

2. How Does the Calculator Work?

The calculator uses polynomial multiplication to expand the factored form:

Where:

• \( a \) — Leading coefficient (defaults to 1 if not specified)
• \( r_1, r_2, \dots, r_n \) — Zeros of the polynomial
• \( (x - r_i) \) — Linear factors corresponding to each zero

Explanation: The calculator multiplies all the linear factors together and then multiplies by the leading coefficient to produce the expanded polynomial form.

3. Importance of Polynomial Construction

Details: Constructing polynomials from zeros is fundamental in algebra, with applications in curve fitting, interpolation, and solving differential equations. The factored form reveals the roots directly.

4. Using the Calculator

Tips: Enter the leading coefficient (typically 1 if not specified) and comma-separated zeros. Complex zeros must come in conjugate pairs for real polynomials.

5. Frequently Asked Questions (FAQ)

Q1: What if my polynomial has complex zeros?
A: The calculator works with real coefficients. For complex zeros, enter them as decimal approximations or ensure conjugate pairs are included.

Q2: How are repeated zeros handled?
A: Simply include the zero multiple times in the input. For a double zero at 3, enter "3, 3".

Q3: What's the maximum degree polynomial this can handle?
A: The calculator can theoretically handle any degree, but display becomes impractical beyond about 10th degree.

Q4: Why is the leading coefficient important?
A: It scales the entire polynomial. Without it, you'd get a monic polynomial (leading coefficient of 1).

Q5: Can this calculator factor polynomials?
A: No, this does the opposite - it expands factored form into standard form. For factoring, you'd need a different tool.