Final answer:
To find the rational root of the cubic polynomial f(x) = 2x^3 + 2x^2 - 7x - 7, one should apply the Rational Root Theorem and test possible values obtained from the factors of the constant term and leading coefficient. The root is found to be x = -1.
Step-by-step explanation:
The function f(x) = 2x^3 + 2x^2 - 7x - 7 is a cubic polynomial, and based on the Rational Root Theorem, any rational root of this polynomial, if it exists, should be of the form ± p/q, where p is a factor of the constant term (in this case 7), and q is a factor of the leading coefficient (in this case 2).
To find the rational root, we should test all possible values of ± p/q, which are ± 1/2, ± 1, ± 7/2, ± 7. By substituting these values into the function and looking for which value of x makes the polynomial equal to zero, we can identify the rational root. After testing these possibilities, you will find that x = -1 is a root, since plugging it into the polynomial yields 0:
f(-1) = 2(-1)^3 + 2(-1)^2 - 7(-1) - 7 = -2 + 2 + 7 - 7 = 0
Therefore, x = -1 is the rational root of the function f(x).