Find a polynomial of least degree with only real coefficients and having the given zeros.
-$3 , - 11$, and $4 + 4 i$
A) $f ( x ) = x ^ { 4 } - 65 x ^ { 2 } + 520 x - 1056$
B) $f ( x ) = x ^ { 4 } - 130 x ^ { 2 } + 520 x - 1056$
C) $f ( x ) = x ^ { 4 } - 7 x ^ { 3 } + 12 x ^ { 2 } - 260 x + 1056$
D) $f ( x ) = x ^ { 4 } - 7 x ^ { 3 } - 12 x ^ { 2 } + 260 x - 1056$

Question

Quizplus · Accepted Answer

Given the zeros $3, -11$, and $4 + 4i$, the polynomial must also have the conjugate of $4 + 4i$, which is $4 - 4i$, to ensure all coefficients are real. The polynomial with these zeros is obtained by multiplying $(x - 3)(x + 11)(x - (4 + 4i))(x - (4 - 4i))$. Simplifying this product leads to a polynomial of degree 4 with real coefficients, matching option A.

Find a Polynomial of Least Degree with Only Real Coefficients 3,−113 , - 113,−11

Find a Polynomial of Least Degree with Only Real Coefficients $3 , - 11$