Solving the Cubic Equation x3 - x2 - x - 1 0

In algebra, solving polynomial equations, especially cubic equations, is a fundamental task. The provided equation x3 - x2 - x - 1 0 is a prime example. This article will walk you through the process of solving this equation using advanced algebraic techniques and the Cardano's formula.

Introduction to the Equation

First, it's important to clarify that the equation presented is not just an expression of an unknown variable but an equation. Equations, unlike expressions, have solutions that we can find. Given the complexity of this specific equation, we need to employ sophisticated methods to determine its roots.

The initial step is to understand that the equation x3 - x2 - x - 1 0 lacks rational roots. This method can be verified through the rational root theorem, which asserts that any potential rational root must be a factor of the constant term (here, -1) divided by the leading coefficient (here, 1). Since there are no such factors that satisfy the equation, we must explore other methods.

Transforming the Equation

To make the equation more tractable, we can perform a substitution. Let x y1/3. This will eliminate the second term when we expand the cubic in terms of y1/3. The equation then becomes:

y3 - 4/3 ? y - 38/27 0

Applying Cardano's Formula

Cardano's formula provides a method to solve depressed cubic equations of the form t3 - pt - q 0. Our transformed equation matches this form, with p -4/3 and q -38/27. The solution can be derived using the formula:

sqrt[3]{-q/2sqrt{q2/4p3/27}} sqrt[3]{-q/2 - sqrt{q2/4p3/27}}

Substituting the values of p and q from our equation, we get:

x ≈ 1.83928675

Exploring Complex Solutions

In addition to the real solution, there are two complex solutions to the equation. These can be found by multiplying the value of u from the complex square roots of unity. This process, while more intricate, is a standard part of solving cubic equations.

The real solution can be written as:

x 1/3 ? sqrt[3]{19 3 sqrt{33}}

Using this, we can derive the two complex solutions as follows:

x 1/3 ? sqrt[3]{19 - 3 sqrt{33}}

These solutions provide a complete picture of the roots of the given cubic equation.

Conclusion

The cubic equation x3 - x2 - x - 1 0 can be solved using advanced algebraic techniques such as substitution and Cardano's formula. The process involves transforming the equation, applying the formula, and then finding both the real and complex solutions. Understanding these methods is crucial for solving more complex polynomial equations.