Is There a General Formula for Finding the Roots of a Cubic Function?

Cubic polynomials, which are a third-degree polynomial of the form (ax^3 bx^2 cx d 0), have notable roots that can be determined using a general formula. This formula, discovered centuries ago, allows us to find the roots without needing specialized software for numerical solutions. In this article, we will explore how to use Cardano's method to find the roots of a cubic equation, along with a brief history and a practical example.

The General Form of a Cubic Equation

A cubic equation is generally written as:

[ax^3 bx^2 cx d 0]

where (a, b, c,) and (d) are constants and (a eq 0). The equation has at least one real root due to the nature of cubic functions.

Depressing the Cubic

The process of “depressing the cubic” involves transforming the cubic equation into a simpler form where the (x^2) term is eliminated. This is achieved by making a substitution:

[displaystyle x y - frac{b}{3a}]

After this substitution, the equation can be rewritten as:

[y^3 py q 0]

where:

[displaystyle p frac{3ac - b^2}{3a^2}] [displaystyle q frac{2b^3 - 9abc 27a^2d}{27a^3}]

Cardano's Formula

Cardano's method, named after Girolamo Cardano, is used to solve the depressed cubic equation. The roots of the depressed cubic equation can be found using the following formulas:

[displaystyle D left(frac{q}{2}right)^2 - left(frac{p}{3}right)^3]

Case 1: (D > 0)

If the discriminant (D) is greater than zero, the cubic equation has three distinct real roots. One of the roots can be found using:

[displaystyle y_1 sqrt[3]{-frac{q}{2} sqrt{D}} sqrt[3]{-frac{q}{2} - sqrt{D}}]

The remaining roots can be derived using the relationships among the roots.

Case 2: (D 0)

When the discriminant (D) equals zero, all roots are real, and at least two are equal. The single distinct root can be found:

[displaystyle y_1 sqrt[3]{-frac{q}{2}}]

The remaining roots can be derived from this value using the relationships among the roots.

Case 3: (D

When the discriminant (D) is less than zero, the cubic equation has one real root and two complex conjugate roots. The real root can be found:

[displaystyle y_1 sqrt[3]{-frac{q}{2} sqrt{D}} sqrt[3]{-frac{q}{2} - sqrt{D}}]

The remaining roots can be found using the complex conjugate property and the relationships among the roots.

Converting Back to the Original Variable

Once the roots (y_1, y_2, y_3) are found, the original roots (x_1, x_2, x_3) can be obtained by undoing the substitution:

[displaystyle x y frac{b}{3a}]

A Brief Example

Consider the cubic equation:

[displaystyle x^3 - 3x^2 3x - 1 0]

Here, (a 1), (b -3), (c 3), and (d -1). Substituting into the formulas, we get:

[displaystyle p frac{3(1)(3) - (-3)^2}{3(1)^2} 0] [displaystyle q frac{2(-3)^3 - 9(1)(-3)(3) 27(1)^2(-1)}{27(1)^3} 0]

Since (p 0) and (q 0), the discriminant (D 0), and the equation has one real root and two equal roots. The real root is:

[displaystyle y_1 sqrt[3]{-frac{0}{2}} 0]

Thus, the original root is:

[displaystyle x 0 frac{-3}{3(1)} 1]

Therefore, the cubic equation (x^3 - 3x^2 3x - 1 0) has the root (x 1) with multiplicity 3.

Conclusion

While finding the roots of a cubic function using the general formula is complex and often performed with the aid of software, the method is a powerful tool in algebra. By employing Cardano's method, we can systematically approach and solve cubic equations, broadening our understanding of polynomial roots.

For those interested in additional resources, the Iranian scientist Musa Nasri discovered a similar formula for solving cubic equations. This historical and academic insight can be further explored for a deeper understanding of the roots of cubic functions.