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) $f ( x ) = - 4 x ^ { 4 } + 22 x ^ { 3 } - 74 x ^ { 2 } + 152 x - 60$

Question

Quizplus · Accepted Answer

Given the zeros $3, \frac{1}{2}, 1 + 3i$, and its conjugate $1 - 3i$ must also be a zero because the coefficients are real. The polynomial can be constructed from its roots as $f(x) = a(x - 3)(x - \frac{1}{2})(x - (1 + 3i))(x - (1 - 3i))$. Expanding and simplifying this expression, then determining the leading coefficient $a$ using $f(1) = -54$, matches option A.

Find an Nth Degree Polynomial Function with Real Coefficients Satisfying n=4;3,12\mathrm { n } = 4 ; 3 , \frac { 1 } { 2 }n=4;3,21​

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