Solving Complex Equations: A Guide to Factorization and Root Extraction

In the realm of advanced mathematics, solving polynomial equations can be both intricate and fascinating. This article delves into the process of solving the equation $x y^6$, which leads to a polynomial of degree 12. We explore the step-by-step factorization of this equation and the identification of its solutions, including both real and complex solutions.

Problem Setup and Equation Transformation

The initial equation is given by $x y^6$, which transforms into a more complex polynomial equation after substitution and simplification:

$y^{12} – 2sqrt{2}y^9 – 4y^8 4y^6 8y^4 0$

By factoring this equation, we aim to simplify the solution process. The polynomial can be factored as:

$y^4(y - sqrt{2}^{2})(y 2sqrt{2})(y^2 6y 4sqrt{2})(y^2 - 6y 4sqrt{2}) 0$

Identifying Solutions

The solutions to this equation can be found by setting each factor equal to zero:

Obvious Solutions

1. $x 0$ 2. $x sqrt{2}^6 8$

Complex Solutions

The degree-6 factor has 3 pairs of complex conjugate solutions, which are:

$x rho_1 pm theta_1 0.882529645955 pm 29.04056572095i$ $x rho_2 pm theta_2 5.048950953552 pm 16.2592747489i$ $x rho_3 pm theta_3 14.3631439066352 pm 10.7806539306i$

These solutions are derived from the polynomial factors and illustrate the presence of complex numbers in the solution set.

Verification of Solutions

To verify the solution for $x 0$, we substitute it into the original equation:

$x (x - 2sqrt{2} cdot x^{1/2} - 4x^{1/3} - frac{8}{x^{1/3}}) x^2 - 2sqrt{2} cdot x^{3/2} - 4 cdot x^{4/3} - frac{8}{x^{1/3}} 0$

When $x 0$, both sides of the equation equal zero, confirming $x 0$ as a valid solution.

General Approach for Solving Equations with Roots

When dealing with equations containing multiple roots, the method of substituting and simplifying can be quite useful. For example, the substitution $x y^6$ can help eliminate some of the roots:

$y^6 - 2sqrt{2}y^3 - 4y^2 - 4frac{8}{y^2} 0$

However, this substitution does not significantly simplify the equation further.

The key to solving such complex equations lies in the factorization of polynomials and the identification of both real and complex roots. Understanding these techniques can greatly enhance your problem-solving skills in advanced mathematics.