Find an nth degree polynomial function with real coefficients satisfying the given conditions.
-$\mathrm { n } = 4 ; 2 \mathrm { i } , 7$, and $- 7$ are zeros; leading coefficient is 1
A) $f ( x ) = x ^ { 4 } - 45 x ^ { 2 } - 196$
B) $f ( x ) = x ^ { 4 } + 4 x ^ { 3 } - 45 x ^ { 2 } - 196$
C) $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } - 196$
D) $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } - 7 x - 196$

Question

Quizplus · Accepted Answer

Complex roots come in conjugate pairs, so if $2i$ is a root, $-2i$ is also a root. The polynomial can be constructed from its roots as $(x - 2i)(x + 2i)(x - 7)(x + 7)$. Simplifying this gives $f(x) = (x^2 + 4)(x^2 - 49)$, which further simplifies to $f(x) = x^4 - 45x^2 - 196$.

Find an Nth Degree Polynomial Function with Real Coefficients Satisfying n=4;2i,7\mathrm { n } = 4 ; 2 \mathrm { i } , 7n=4;2i,7

Find an Nth Degree Polynomial Function with Real Coefficients Satisfying $\mathrm { n } = 4 ; 2 \mathrm { i } , 7$