Question

X^4+4x^3-18x^2-12x+9=0

Answer 1

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Answer 2

To solve the equation x^4 + 4x^3 - 18x^2 - 12x + 9 = 0, we can start by trying to factor it or find its roots. Unfortunately, this polynomial does not appear to have any simple rational roots. Therefore, we may need to resort to numerical methods or a computer algebra system to find its roots. One such numerical method is the Newton-Raphson method.

Using the Newton-Raphson method, we can iteratively approximate the roots of the polynomial. Here's how you can do it:

1. Start with an initial guess for the root, let's say x_0.

2. Use the following iteration formula to update your guess:

   x_(n+1) = x_n - f(x_n) / f'(x_n)

   where f(x) is the polynomial and f'(x) is its derivative.

3. Keep iterating until you reach an approximation that is sufficiently close to the actual root.

Let's start with an initial guess of x_0 = 1 and use the Newton-Raphson method to find one of the roots:

f(x) = x^4 + 4x^3 - 18x^2 - 12x + 9

f'(x) = 4x^3 + 12x^2 - 36x - 12

Now, we can apply the Newton-Raphson method:

1st Iteration: x_1 = x_0 - [f(x_0) / f'(x_0)] x_1 = 1 - [(1^4 + 41^3 - 181^2 - 121 + 9) / (41^3 + 121^2 - 361 - 12)] x_1 ≈ 1.2727

2nd Iteration: x_2 = x_1 - [f(x_1) / f'(x_1)] x_2 ≈ 1.2727 - [(1.2727^4 + 41.2727^3 - 181.2727^2 - 121.2727 + 9) / (41.2727^3 + 121.2727^2 - 361.2727 - 12)] x_2 ≈ 1.1676

3rd Iteration: x_3 = x_2 - [f(x_2) / f'(x_2)] x_3 ≈ 1.1676 - [(1.1676^4 + 41.1676^3 - 181.1676^2 - 121.1676 + 9) / (41.1676^3 + 121.1676^2 - 361.1676 - 12)] x_3 ≈ 1.1545

Continue this process until you reach an approximation that doesn't change much between iterations. This will give you an approximate root of the equation. Repeat the process with different initial guesses to find all possible real roots of the polynomial. Keep in mind that this polynomial may have complex roots as well.