How to find all roots of"x³-7x²+11x-5=0"

cubics are not easily done in general, but a little synthetic division should quickly show you that you have

(x-1)(x^2-6x+5)

and that is easy...

Very nice

To find all the roots of the equation x³ - 7x² + 11x - 5 = 0, you can use a method called factoring or synthetic division. However, the given equation is a cubic equation, and factoring may not be a straightforward option. To solve this equation algebraically, you can use a numerical method called the Newton-Raphson method. Here's how you can find the roots using this method:

1. Begin by rearranging the equation to the form f(x) = 0: x³ - 7x² + 11x - 5 = 0.

2. Calculate the derivative of f(x) with respect to x. In this case, the derivative of f(x) is f'(x) = 3x² - 14x + 11.

3. Choose an initial guess for one of the roots. Let's start with x = 1 as an initial guess.

4. Use the Newton-Raphson method formula to refine the initial guess and find a more accurate solution. The formula is given by:

xᵢ₊₁ = xᵢ - (f(xᵢ) / f'(xᵢ))

In this equation, xᵢ represents the ith approximation of the root, and xᵢ₊₁ represents the (i+1)th approximation of the root.

5. Plug in the values of f(x) and f'(x) into the formula. For the given equation, f(x) = x³ - 7x² + 11x - 5 and f'(x) = 3x² - 14x + 11.

6. Iterate the formula using the previous result as the new approximation until the result converges to a specific value. Repeat the iterations until the approximation becomes accurate enough.

7. Repeat steps 3-6 using different initial guesses to find all the roots of the equation.

By following these steps, you can find the roots of the given equation using the Newton-Raphson method.