Final answer:
The stability of a polynomial is determined by the nature of its roots, which for a quadratic equation, can be found using the quadratic formula. The discriminant indicates whether the roots are real or complex. Without the correct coefficients, stability cannot be assessed.
Step-by-step explanation:
To assess the stability of a polynomial, we need to understand whether its solutions are real and non-negative, which is particularly relevant in fields like control theory. However, the polynomial presented in the question seems to be incomplete or incorrect. Let's consider the standard form of a quadratic equation, which is at² + bt + c = 0. In this form, a, b, and c are constants, and the solutions are found using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
The discriminant b² - 4ac determines the nature of the roots:
- If it is greater than zero, the equation has two distinct real roots.
- If it is equal to zero, the equation has one real root (also known as a repeated or double root).
- If it is less than zero, the equation has two complex roots, which are not considered stable in the context of real-life systems.
Without the correct form of the polynomial and its coefficients, it is impossible to determine the stability. If the student can provide the correct polynomial with specific coefficients, then we can apply the quadratic formula to determine the stability of its roots.