Answer:
According to the Rational Root Theorem, the possible rational roots of a polynomial can be found by considering the factors of the constant term (in this case, 64) divided by the factors of the leading coefficient (in this case, 3).
To find the rational roots of the polynomial 3x^3 - 16x^2 - 12x + 64, we need to consider the factors of 64 and the factors of 3.
The factors of 64 are:
1, 2, 4, 8, 16, 32, 64
The factors of 3 are:
1, 3
Now, we can form possible rational roots by taking combinations of the factors. For example, dividing any of the factors of 64 by any of the factors of 3:
Possible rational roots:
±1/1, ±2/1, ±4/1, ±8/1, ±16/1, ±32/1, ±64/1
±1/3, ±2/3, ±4/3, ±8/3, ±16/3, ±32/3, ±64/3
These possible rational roots can be used to test for actual roots of the polynomial. By substituting each possible root into the polynomial and checking if the result is zero, we can determine which of these roots are indeed rational roots of the given polynomial.
Note: The Rational Root Theorem provides a list of possible rational roots, but it does not guarantee that all or any of these roots are actual roots of the polynomial.
Step-by-step explanation:
