Find an Nth Degree Polynomial Function with Real Coefficients Satisfying $\mathrm { n } = 4 ; 3 , \frac { 1 } { 2 }$

Multiple Choice

Find an nth degree polynomial function with real coefficients satisfying the given conditions.
- $\mathrm { n } = 4 ; 3 , \frac { 1 } { 2 }$ , and $1 + 3 \mathrm { i }$ are zeros; $\mathrm { f } ( 1 ) = - 54$

A) $f ( x ) = 6 x ^ { 4 } - 33 x ^ { 3 } + 111 x ^ { 2 } - 228 x + 90$
B) $f ( x ) = 2 x ^ { 4 } - 11 x ^ { 3 } + 37 x ^ { 2 } - 228 x + 90$
C) $f ( x ) = 3 x ^ { 4 } - 33 x ^ { 3 } + 111 x ^ { 2 } - 228 x + 90$
D) $f ( x ) = - 4 x ^ { 4 } + 22 x ^ { 3 } - 74 x ^ { 2 } + 152 x - 60$