Remaining Roots Calculator

This calculator helps you find the remaining roots of a polynomial after one root is known. It's particularly useful in solving polynomial equations where you already have one solution and need to find the others.

What is Remaining Roots?

When solving polynomial equations, it's often helpful to find all roots of the equation. If you already know one root, you can use polynomial division to find the remaining roots. This process is called "factoring out" the known root.

The remaining roots calculator automates this process, making it easier to find all solutions to a polynomial equation.

Note: This calculator works best with polynomials where you know at least one real root. For complex roots, additional mathematical techniques may be required.

How to Use the Calculator

Enter the coefficients of your polynomial in the order from highest degree to lowest (e.g., for 3x³ + 2x² - 5x + 1, enter 3, 2, -5, 1).
Enter the known root of the polynomial.
Click "Calculate" to find the remaining roots.
Review the results and chart visualization if available.

Formula

To find the remaining roots after knowing one root (r), we use polynomial division:

Given polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

If r is a root, then P(x) = (x - r)Q(x)

Where Q(x) is the quotient polynomial representing the remaining roots.

The calculator performs this division and solves Q(x) = 0 to find the remaining roots.

Example Calculation

Let's find the remaining roots of the polynomial 2x³ - 3x² - 11x + 6 with a known root of 3.

First, perform polynomial division of (2x³ - 3x² - 11x + 6) by (x - 3).
The quotient polynomial will be x² + 0x - 2 (or x² - 2).
Set the quotient equal to zero: x² - 2 = 0.
Solve for x: x = ±√2.

The remaining roots are √2 and -√2.

FAQ

What if I don't know any roots of the polynomial?

You can use other methods like the Rational Root Theorem or graphing to estimate possible roots before using this calculator.

Can this calculator handle complex roots?

Yes, but complex roots may require additional mathematical interpretation beyond what this calculator provides.

What if the polynomial has repeated roots?

The calculator will show the multiplicity of each root in the results.