Use Roots to Find Polynomial Calculator

How to Use Roots to Find a Polynomial

The process of finding a polynomial from its roots involves creating a product of binomials, each representing a root. Here's a step-by-step method:

1. Identify all the roots of the polynomial. These are the values of x that make the polynomial equal to zero.
2. For each root, create a binomial factor of the form (x - r), where r is the root.
3. Multiply all the binomial factors together to form the polynomial.
4. If there are repeated roots, include the appropriate power of each binomial factor.
5. Simplify the expression if possible.

This method works for polynomials with real and complex roots. The resulting polynomial will have the given roots and will be of degree equal to the number of roots (counting multiplicities).

Formula for Finding a Polynomial from Roots

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

Where:

• P(x) is the polynomial
• a is a constant coefficient (often 1 if not specified)
• r₁, r₂, ..., rₙ are the roots of the polynomial

For repeated roots, the binomial factor is raised to the power equal to the multiplicity of the root. For example, if root r has multiplicity m, the factor becomes (x - r)ᵐ.

Worked Example

Let's find the polynomial with roots at x = 2, x = -1, and x = 3.

1. Create binomial factors for each root:
   • (x - 2)
   • (x - (-1)) = (x + 1)
   • (x - 3)
2. Multiply the factors together:
   P(x) = (x - 2)(x + 1)(x - 3)
3. Expand the expression:
   P(x) = (x² - x - 2)(x - 3) P(x) = x³ - 3x² - x² + 3x - 2x + 6 P(x) = x³ - 4x² + x + 6

The resulting polynomial is x³ - 4x² + x + 6, which has roots at x = 2, x = -1, and x = 3.