Final answer:
The Rational Zero Theorem is used to list potential rational zeros for a polynomial function with integer coefficients. For the function f(x) = -3x^4 + 5x^3 + 2x^2 + 2x + 6, the theorem identifies possible zeros based on the factors of the constant term and the leading coefficient. Test these potential zeros using synthetic division or other suitable methods.
Step-by-step explanation:
The Rational Zero Theorem
The Rational Zero Theorem provides a list of potential rational zeros for a polynomial function. For a polynomial function f(x) with integer coefficients, if there is a rational zero p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p can be any factor of the constant term and q can be any factor of the leading coefficient. Applying the Rational Zero Theorem to the given function f(x) = -3x^4 + 5x^3 + 2x^2 + 2x + 6, we need to find factors of the constant term (6) and factors of the leading coefficient (-3).
The factors of 6 are ±1, ±2, ±3, and ±6, while the factors of -3 are ±1 and ±3. Therefore, the potential rational zeros of the function could be ±1/1, ±2/1, ±3/1, ±6/1, -1/1, -2/1, -3/1, -6/1, ±1/3, ±2/3, ±3/3, ±6/3, -1/3, -2/3, -3/3, and -6/3. These values simplify to ±1, ±2, ±3, ±6, -1, -2, -3, -6, ±1/3, ±2/3, 1, 2, -1/3, -2/3, -1, and -2, respectively. After identifying these potential rational zeros, one would usually use synthetic division or another method to test which, if any, are actual zeros of the function.