Polynomial Calculator From Zeros

Introduction & Importance

Polynomials are fundamental in algebra, with roots (zeros) determining their behavior. Our calculator simplifies finding coefficients from given roots, aiding in polynomial construction and analysis.

How to Use This Calculator

1. Enter roots (zeros) separated by commas in the ‘Zeros’ field.
2. Click ‘Calculate’.
3. View results in the ‘Results’ box and the chart.

Formula & Methodology

The formula for a polynomial with roots (zeros) r₁, r₂, …, r_n is:

P(x) = a(x – r₁)(x – r₂)…(x – r_n)

where a is the leading coefficient, calculated as:

a = (-1)^n * r₁ * r₂ * … * r_n

Real-World Examples

Example 1: Quadratic Polynomial

Roots: -2, 3

Coefficients: -6, 11, -6

Example 2: Cubic Polynomial

Roots: 1, 2, 3

Coefficients: -6, 11, -12, 6

Example 3: Quartic Polynomial

Roots: -1, 1, 2, 3

Coefficients: 1, -6, 11, -6, 1

Data & Statistics

Degree	Number of Roots	Polynomial Form
1	1	ax + b
2	2	ax^2 + bx + c
3	3	ax^3 + bx^2 + cx + d
4	4	ax^4 + bx^3 + cx^2 + dx + e

Expert Tips

• Entering roots in decreasing order can simplify the calculation.
• For complex roots, use the calculator to find coefficients, then apply the conjugate root theorem.

Interactive FAQ

What are the roots of a polynomial?

Roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero.

How many roots can a polynomial have?

A polynomial of degree n can have up to n roots, counting multiplicity.

References

Math is Fun: Polynomials

Khan Academy: Algebra Basics