Use the remainder theorem to determine if the given number \( c \) is a zero of the polynomial. \[ f(x)=3 x^{3}-14 x^{2}+111 x+180 \] (a) \( c=3+i \) (b) \( c=3-i \)

To determine if \( c = 3 + i \) is a zero of the polynomial \( f(x) = 3x^3 - 14x^2 + 111x + 180 \), we can evaluate \( f(3 + i) \). Plugging this complex number into the polynomial might sound tricky, but it’s just substituting and simplifying. If \( f(3 + i) = 0 \), then \( 3 + i \) is a zero. Now, let's try \( c = 3 - i \). Interestingly, if \( f(3 + i) \) results in a non-zero value, we can use the fact that complex zeros come in conjugate pairs. So, evaluating \( f(3 - i) \) would yield similar results. If one is a zero, typically the other is too! Grab your calculator or paper, and dive in to see what values pop out!