Final answer:
To apply the Rational Roots Theorem to the polynomial f(x), multiply by 3 to get integer coefficients and then test the factors of 3 divided by factors of 3 (±1, ±3) as potential roots. Verify the roots by substituting them in, and if a root is found, use polynomial division to simplify further and solve any resulting quadratic equation.
Step-by-step explanation:
To use the Rational Roots Theorem (RRT) to find rational roots of the polynomial f(x) = x3 + (7/3)x2 - (11/3)x + 1, we first need to multiply through by 3 to eliminate the fractions and have integer coefficients. This gives us the polynomial 3f(x) = 3x3 + 7x2 - 11x + 3. Now, according to RRT, the possible rational roots are the factors of the constant term (3) divided by the factors of the leading coefficient (3), which in this case are ±1, ±3.
Next, we can test these possible roots by substituting them into the polynomial and looking for any values that yield zero. If we find such values, we can deem them as roots of the polynomial. After going through the list of possible roots, we can conclude which of them are actual roots. Remember, when testing these possible roots in 3f(x), the roots found will also be the roots of f(x) since we are merely scaling the polynomial and not altering its roots. Additionally, always check the answer to see if it makes sense in the context of the given polynomial.
Eliminate terms and simplify the algebra through synthetic division or the Remainder Theorem when testing the possible roots. If a root is indeed valid, polynomial division will further reduce the polynomial, possibly transforming it into a quadratic, for which you can use the quadratic formula to find any remaining roots.