Cal11 calculator
Search calculator...

Roots of A Polynomial Given One Root Calculator

Reviewed by Calculator Editorial Team

Finding all roots of a polynomial when you know one root is a common problem in algebra. This calculator helps you determine the other roots by factoring the polynomial based on the given root.

Introduction

When you know one root of a polynomial, you can use polynomial division to factor the polynomial and find the other roots. This process is particularly useful when dealing with higher-degree polynomials where finding all roots analytically might be difficult.

The calculator on this page uses the Factor Theorem to determine the other roots. The Factor Theorem states that if r is a root of a polynomial P(x), then (x - r) is a factor of P(x).

How to Use This Calculator

  1. Enter the coefficients of your polynomial in the order from highest degree to lowest degree.
  2. Enter the known root of the polynomial.
  3. Click "Calculate" to find the other roots.
  4. Review the results and chart visualization.

Note: The calculator assumes the polynomial is factorable based on the given root. For polynomials with irrational or complex roots, the calculator may not provide exact solutions.

Formula

The calculator uses polynomial division to factor the polynomial based on the given root. The general approach is:

  1. Given polynomial P(x) = anxn + an-1xn-1 + ... + a0
  2. Given root r, factor out (x - r) from P(x)
  3. Perform polynomial division to find the quotient polynomial
  4. Find the roots of the quotient polynomial
  5. Combine with the known root to get all roots

If P(r) = 0, then P(x) = (x - r)Q(x), where Q(x) is the quotient polynomial.

Worked Example

Let's find all roots of the polynomial P(x) = x3 - 6x2 + 11x - 6 given that x = 1 is a root.

  1. Verify that P(1) = 1 - 6 + 11 - 6 = 0, confirming x = 1 is a root.
  2. Factor out (x - 1) from P(x):

    P(x) = (x - 1)(x2 - 5x + 6)

  3. Find roots of the quadratic factor x2 - 5x + 6:

    x = [5 ± √(25 - 24)]/2 = [5 ± 1]/2

    x = 3 or x = 2

  4. All roots are x = 1, x = 2, x = 3.

FAQ

What if the polynomial has complex roots?
The calculator will provide the complex roots in the form a + bi where i is the imaginary unit.
Can I use this calculator for polynomials with non-integer coefficients?
Yes, the calculator accepts any real number coefficients and roots.
How accurate are the results?
The calculator uses precise polynomial division and root-finding algorithms to provide accurate results.
What if I don't know any roots of the polynomial?
You can use other root-finding methods like the Rational Root Theorem or graphing to estimate possible roots.