Root Calculator Javascript Polynomial

How to Use This Calculator

To find the roots of a polynomial equation using this calculator:

1. Enter the coefficients of your polynomial in the input fields. For example, for the polynomial 2x³ - 5x² + 3x - 7, you would enter 2 for x³, -5 for x², 3 for x, and -7 for the constant term.
2. Select the method you want to use for finding the roots (Newton-Raphson, Secant, or Bisection).
3. Click the "Calculate" button to find the roots.
4. Review the results, which will show the roots of the polynomial equation.

The calculator will display the roots of the polynomial equation and a chart visualizing the polynomial function and its roots.

Understanding Polynomial Roots

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation. The roots of a polynomial are the values of the variable that make the polynomial equal to zero.

For example, the roots of the polynomial x² - 5x + 6 are 2 and 3, because (2)² - 5(2) + 6 = 0 and (3)² - 5(3) + 6 = 0.

Finding the roots of a polynomial is a fundamental problem in algebra and numerical analysis. There are several methods for finding the roots of a polynomial, including:

• Newton-Raphson method
• Secant method
• Bisection method
• Factorization

JavaScript Methods for Finding Roots

JavaScript provides several methods for finding the roots of polynomial equations. The most common methods are the Newton-Raphson method, the Secant method, and the Bisection method.

Newton-Raphson Method

The Newton-Raphson method is an iterative numerical method for finding successively better approximations to the roots of a real-valued function. The method uses the idea of linear approximation (using the tangent line) to find the root of a function.

Secant Method

The Secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate the root of a function. The method is similar to the Newton-Raphson method but does not require the derivative of the function.

Bisection Method

The Bisection method is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. The method is guaranteed to converge to a root if the function is continuous and the interval contains a root.

Worked Example

Let's find the roots of the polynomial 2x³ - 5x² + 3x - 7 using the Newton-Raphson method.

1. First, we need to find the derivative of the polynomial. The derivative of 2x³ - 5x² + 3x - 7 is 6x² - 10x + 3.
2. We'll use an initial guess of x₀ = 2.
3. Calculate f(x₀) = 2(2)³ - 5(2)² + 3(2) - 7 = 16 - 20 + 6 - 7 = -5.
4. Calculate f'(x₀) = 6(2)² - 10(2) + 3 = 24 - 20 + 3 = 7.
5. Calculate the next approximation: x₁ = x₀ - f(x₀)/f'(x₀) = 2 - (-5)/7 ≈ 2.714.
6. Repeat the process with x₁ to find x₂, and so on until the approximation converges to a root.

After several iterations, we find that the roots of the polynomial are approximately 2.714, 1.286, and -1.000.