Form a polynomial f(x)with real coefficients having the given degree and zeros.
-Degree: 5; zeros: $2 , - 3 i$, and $4 - i$
A) $f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 26 x ^ { 3 } - 124 x ^ { 2 } - 72 x - 306$
B) $f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 42 x ^ { 3 } - 124 x ^ { 2 } + 297 x - 306$
C) $f ( x ) = x ^ { 5 } - 10 x ^ { 4 } - 42 x ^ { 3 } - 124 x ^ { 2 } + 297 x + 306$
D) $f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 26 x ^ { 3 } - 124 x ^ { 2 } + 72 x + 306$

Question

Quizplus · Accepted Answer

Given the zeros $2, -3i$, and $4-i$, we know that if a polynomial has real coefficients, the complex zeros must come in conjugate pairs. Therefore, the zeros of the polynomial are $2, -3i, 3i, 4-i$, and $4+i$. The factors corresponding to these zeros are $(x-2), (x+3i), (x-3i), (x-(4-i))$, and $(x-(4+i))$. Multiplying these factors together gives a polynomial of degree 5 with real coefficients. The correct polynomial that fits these criteria is option B.

Form a Polynomial F(x)with Real Coefficients Having the Given Degree 2,−3i2 , - 3 i2,−3i

Form a Polynomial F(x)with Real Coefficients Having the Given Degree $2 , - 3 i$