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more than 4 weeks ago

To find the roots of the equation f(x) = 0, we need to solve the equation x^3 + 6x^2 + x - 4 = 0.

One way to find the roots is by factoring the polynomial. However, in this case, the polynomial is not easily factorable. Therefore, we can use numerical methods such as the Newton-Raphson method or synthetic division to approximate the roots.

Using the Newton-Raphson method, we start with an initial guess for the root and iterate using the formula:

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

where f'(x) is the derivative of the function f(x).

By iterating this formula, we can find the roots of the equation f(x) = 0.

One way to find the roots is by factoring the polynomial. However, in this case, the polynomial is not easily factorable. Therefore, we can use numerical methods such as the Newton-Raphson method or synthetic division to approximate the roots.

Using the Newton-Raphson method, we start with an initial guess for the root and iterate using the formula:

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

where f'(x) is the derivative of the function f(x).

By iterating this formula, we can find the roots of the equation f(x) = 0.

Google Gemini AI

more than 4 weeks ago

- List the factors of the constant term (-4): ±1, ±2, ±4

- List the factors of the leading coefficient (1): ±1

- Try all possible combinations of the factors to find rational roots.

We find that

```

2 | 1 6 1 -4

\ 2 8 18

---

1 8 9 14

```

The last number in the bottom row (14) is the remainder, which is not zero. Therefore,

Since 2 is not a root, the numerator of the rational root must be a factor of 14. The factors of 14 are: ±1, ±2, ±7, ±14

Repeat the synthetic division process for each factor of 14. We find that

Using synthetic division, we divide f(x) by (x - (-7)):

```

-7 | 1 6 1 -4

\ -7 35 -245

---

1 -1 36 -249

```

The reduced polynomial is

Using the quadratic formula, we can find the roots of x^2 - x + 36 = 0:

```

x = (1 ± √(1 - 4 - 1 - 36)) / 2 - 1

x = (1 ± √(-143)) / 2

```

Since there are no real solutions to the quadratic, the roots of f(x) = 0 are: