Derive the relationship between the coefficients and the roots of the cubic equation \( x^3 + px^2 + qx + r = 0 \).
user
Sure, here's the relationship between the coefficients and the roots of the cubic equation:

Let's assume the cubic equation is of the form:

```
x³ + ax² + bx + c = 0
```

Where a, b, and c are the coefficients of the equation.

The solution for this equation is called a cubic root or a cube root. The roots of this equation will be one of the following five cases:

1. Three real roots (x1, x2, x3)
2. One real root and two complex conjugates (x + iy, x - Iy)
3. Three equal real roots (x1 = x2 = x3)
4. One real root and two symmetric real roots (x1 = x + Iy, x2 = x - Iy)
5. All three real roots are symmetric (x1 = x2 = -x3)

To find the relationship between the coefficients and the roots of the cubic equation, we can apply Cardano's formula, named after the 16th-century Italian mathematician Gerolamo Cardano.

Cardano's formula is:

```
x = qQ + r, where
Q = (-b ± √(3√3(s-r^3))) / 2π, and
R = (b ± √(3√3(s-r^3))) / (6πa)
```

where s = a³ + b²/2 + c.

The minus or plus sign depends on which root you want to find.

This formula can be quite complicated to compute by hand, so it's usually better to use a calculator or computer software. However, understanding the formula is still useful for a deeper understanding of the relationships.
assistant
Derive the relationship between the coefficients and the roots of the cubic equation \( x^3 + px^2 + qx + r = 0 \).
user
For a cubic equation of the form \( x^3 + px^2 + qx + r = 0 \), let's denote its roots by \( \alpha, \beta, \gamma \). The relationship between the coefficients and roots can be derived using Vieta's formulas.

Vieta's formulas for cubic equations state that:

1. Sum of the roots: \( \alpha + \beta + \gamma = -p \)
2. Sum of the pairwise products of the roots: \( \alpha\beta + \beta\gamma + \gamma\alpha = q \)
3. Product of the roots: \( \alpha\beta\gamma = -r \)

In summary, for the given cubic equation:
- The coefficient \( p \) is the negative sum of the roots.
- The coefficient \( q \) is the pairwise product of the roots.
- The coefficient \( r \) is the negative product of all the roots.
assistant