Write a polynomial f (x) that meets the given conditions.
-Polynomial of lowest degree with real coefficients and with zeros $- 8 + 4 i$ (multiplicity 1 ) and 0 (multiplicity 2)
A) $f ( x ) = x ^ { 4 } + 16 x + 48$
B) $f ( x ) = x ^ { 4 } + 16 x ^ { 3 } + 80 x ^ { 2 }$
C) $f ( x ) = x ^ { 4 } + 16 x + 80$
D) $f ( x ) = x ^ { 4 } + 16 x ^ { 3 } + 48 x ^ { 2 }$

Question

Quizplus · Accepted Answer

Given a complex zero $-8 + 4i$, its conjugate $-8 - 4i$ must also be a zero due to the polynomial having real coefficients. The zeros are $-8 + 4i$, $-8 - 4i$, and $0$ with multiplicity 2. The polynomial can be constructed as $f(x) = (x - 0)^2(x - (-8 + 4i))(x - (-8 - 4i))$, simplifying to $f(x) = x^2(x + 8 - 4i)(x + 8 + 4i)$. Expanding and simplifying this expression, considering the real coefficients and the given zeros, leads to a polynomial of degree 4, matching option B's structure and coefficients when fully expanded.

Write a Polynomial F (X) That Meets the Given Conditions −8+4i- 8 + 4 i−8+4i

Write a Polynomial F (X) That Meets the Given Conditions $- 8 + 4 i$