Final answer:
The equation X²Y²−Z²(X²+Y²)=0 is not a rational equation as it is not a ratio of two polynomials, but it is a polynomial equation which can be factored and solved.
Step-by-step explanation:
The question is asking if the equation X²Y²−Z²(X²+Y²)=0 is rational. To address this, we need to understand what a rational equation is. An equation is considered rational if it can be expressed as the ratio of two polynomials. In this case, we can see that the given equation is a polynomial equation, not a ratio of two polynomials. Hence, the equation is not rational in the strict sense that it is not a ratio of two polynomials. However, it is a valid algebraic equation which can be factored and solved. To illustrate, let's factor the left-hand side of the equation:
X²Y²−Z²(X²+Y²)
We can factor out a common term X²Y², and the equation becomes:
X²Y²(1 − Z²) = 0
This simplifies to either X²Y² = 0 or 1 − Z² = 0. Thus, the solutions can be found when either X or Y is zero, or when Z is ± 1. In summary, while the equation is not rational in the typical definition, it is a solvable algebraic equation.