(a) Determine if the upper bound theorem identifies 4 as an upper bound for the real zeros of \( f(x) \). (b) Determine if the lower bound theorem identifies -1 as a lower bound for the real zeros of \( f(x) \). \( f(x)=2 x^{3}+12 x^{2}+4 x+24 \)

To determine if 4 is an upper bound for the real zeros of \( f(x) = 2x^3 + 12x^2 + 4x + 24 \) using synthetic division, weperform synthetic division with 4. If all the coefficients of the resulting polynomial are non-negative, then 4 is an upper bound. Performing synthetic division with 4 gives the coefficients of the resulting polynomial as non-negative numbers: (2, 12, 4, 24) implies 4 is indeed an upper bound. For -1 as a lower bound, we apply synthetic division with -1. If the coefficients of the resulting polynomial are non-negative (or alternate in sign starting with non-negative), then -1 is a lower bound. The synthetic division with -1 yields coefficients that do not satisfy the condition of being all non-negative, thus -1 is not a lower bound according to the theorem.