To determine whether the given functions are positive real functions (PRF), we need to check two conditions:
1. The numerator of the function should have all real roots, and
2. The denominator of the function should not have any real roots.
Let's analyze each function separately:
1. F(s) = (s^3 + 6s^2 + 2s + 1) / (s + 1)^2
a. Numerator:
The numerator is a polynomial function of degree 3, and it is not factorable easily. To determine whether it has real roots, we can use the Rational Root Theorem or a graphing calculator. After evaluating the function, we find that the roots are not real. Therefore, the numerator of this function does not have any real roots.
b. Denominator:
The denominator is (s + 1)^2, which is a quadratic function. Since it has only one real root (-1) and it is raised to an even power, the denominator does not have any real roots.
Since both the numerator and denominator of F(s) do not have any real roots, F(s) can be classified as a positive real function.
2. F(s) = (s^3 + 2s^2 + 3s + 1) / (s^3 + 5s^2 + 11s + 10)
a. Numerator:
Similar to the previous function, the numerator is a polynomial function of degree 3. By evaluating the function, we find that it has real roots. Therefore, the numerator of this function has real roots.
b. Denominator:
The denominator is also a polynomial function of degree 3. By evaluating the function, we find that it has real roots. Therefore, the denominator of this function also has real roots.
Since both the numerator and denominator of F(s) have real roots, F(s) cannot be classified as a positive real function.
In conclusion, the first function F(s) = (s^3 + 6s^2 + 2s + 1) / (s + 1)^2 is a positive real function, while the second function F(s) = (s^3 + 2s^2 + 3s + 1) / (s^3 + 5s^2 + 11s + 10) is not a positive real function.