Find a polynomial function of the lowest degree with real coefficients that has the zeros below and whose leading coefficient is one.
$0,2,4 - i$
A) $f ( x ) = x ^ { 4 } - 34 x ^ { 3 } + 32 x ^ { 2 } - 10 x$
B) $f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - 34 x$
C) $f ( x ) = x ^ { 4 } + 32 x ^ { 3 } - 10 x ^ { 2 } - 34 x$
D) $f ( x ) = x ^ { 4 } - 10 x ^ { 3 } - 34 x ^ { 2 } + 32 x$
E) $f ( x ) = x ^ { 4 } + 32 x ^ { 3 } - 34 x ^ { 2 } - 10 x$

Question

Quizplus · Accepted Answer

Given the zeros $0, 2,$ and $4 - i$, we also need to include the complex conjugate $4 + i$ to ensure real coefficients. The polynomial can be constructed from these roots as follows: $(x)(x-2)(x-(4-i))(x-(4+i))$. Expanding and simplifying this expression, we obtain a polynomial of degree 4 with a leading coefficient of 1. The correct polynomial, after performing the necessary algebra, matches the option B: $f(x) = x^4 - 10x^3 + 33x^2 - 34x$.

Find a Polynomial Function of the Lowest Degree with Real 0,2,4−i0,2,4 - i0,2,4−i

Find a Polynomial Function of the Lowest Degree with Real $0,2,4 - i$