Use the rational zeros theorem to list the potential rational zeros of the polynomial function. Do not attempt to find the zeros. \[ f(x)=2 x^{7}-x^{5}+8 x^{4}+12 \] Find the potential rational zeros. Choose the correct answer below. A. \( -1,1,-2,2,-4,4,-3,3,-12,12,-\frac{1}{2}, \frac{1}{2},-\frac{1}{12}, \frac{1}{12} \) B. \( -1,1,-2,2,-4,4,-3,3,-6,6,-12,12,-\frac{1}{2}, \frac{1}{2},-\frac{1}{12}, \frac{1}{12} \) C. \( -1,1,-2,2,-4,4,-3,3,-6,6,-12,12,-\frac{1}{2}, \frac{1}{2},-\frac{3}{2}, \frac{3}{2} \) D. \( -1,1,-2,2,-4,4,-3,3,-12,12,-\frac{1}{2}, \frac{1}{2},-\frac{3}{2}, \frac{3}{2},-\frac{1}{12}, \frac{1}{12} \)

The Rational Zeros Theorem states that any rational zero of a polynomial of the form   f(x) = aₙxⁿ + ... + a₀ must be of the form p/q, where p divides the constant term a₀ and q divides the leading coefficient aₙ. For the polynomial   f(x) = 2x⁷ - x⁵ + 8x⁴ + 12, the constant term is 12 and the leading coefficient is 2. Step 1: List factors of the constant term (12):   p ∈ {±1, ±2, ±3, ±4, ±6, ±12}. Step 2: List factors of the leading coefficient (2):   q ∈ {±1, ±2}. Step 3: Form all fractions p/q and simplify: • When q = 1:   The candidates are ±1, ±2, ±3, ±4, ±6, ±12. • When q = 2:   The candidates are ±1/2, ±2/2, ±3/2, ±4/2, ±6/2, ±12/2.   Simplify:    ±2/2 = ±1 (already listed),    ±4/2 = ±2 (already listed),    ±6/2 = ±3 (already listed),    ±12/2 = ±6 (already listed).   So, the new candidate from this set is ±3/2. Thus, the complete list of potential rational zeros is:   ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2. Looking at the multiple-choice answers, option C matches this list.