The process to find all the possible rational roots of a polynomial is determined by the Rational Root Theorem. In this case, we have the 4th degree polynomial equation x^4 - 6x^3 - 5x^2 + 24x + 4 = 0, where 1 is the coefficient of the leading term (x^4) and 4 is the constant term. The Rational Root Theorem states that any potential rational root of the equation can be expressed in the form of p/q where:
1. p is a factor of the constant term (4 in our equation), and
2. q is a factor of the leading coefficient (1 in our equation).
To find all the possible rational roots, we first identify all the factors of 4, which are ±1, ±2, ±4. We then find all the factors of 1, which are ±1.
Now, all we have to do is derive all the possible combinations of p/q, where p is a factor of 4 and q is a factor of 1. Since q has only one value, i.e., 1, all factors of 4 will be our possible rational roots.
After evaluating, we find that the possible rational roots of our given polynomial equation are:
1.0, -1.0, 2.0, -2.0, 4.0, -4.0
These are all the potential solutions to the mentioned polynomial equation according to the Rational Root Theorem. Please, keep in mind that these are just possible solutions, to find the actual roots of the equation, we should substitute these values into the original equation and see if they satisfy it.