Find the roots of the equation f(x) = x3 - 0.2589x2 + 0.02262x -0.001122 = 0

The root of the equation is x ≈ 0.162035

Step-by-step explanation:

To find the roots of the equation you can use the Newton-Raphson method.

It is a way to find a good approximation for the root of a real-valued function f(x) = 0. The method starts with a function f(x) defined over the real numbers, the function derivative f', and an initial guess for a root of the function. It uses the idea that a continuous and differentiable function can be approximated by a straight line tangent to it.

This is the expression that we need to use

For the information given:

For the initial value you can choose although you can choose any value that you want.

So for approximation

Next, with you put it into the equation

, you can see that this value is close to 0 but we need to refine our solution.

For approximation

Again we put into the equation

this value is close to 0 but again we need to refine our solution.

We can summarize this process in the following table.

The approximation gives you the root of the equation.

When you plot the equation you find that only have one real root and is approximate to the value found.

b

step-by-step explanation: