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

Since the polynomial has real coefficients, the complex conjugate of $4-i$ must also be a zero. Therefore, the zeros are $0, 2, 4-i,$ and $4+i$. The polynomial of lowest degree with leading coefficient $1$ and these zeros is $(x-0)(x-2)(x-(4-i))(x-(4+i))$. Expanding this, we get $x^4-10x^3+33x^2-34x$, which matches answer choice B.

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 $0,2,4 - i$