Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the
polynomial in standard form.
-$5 i$ and $\sqrt { 2 }$
A) $f ( x ) = x ^ { 4 } - 23 x ^ { 2 } - 50$
B) $f ( x ) = x ^ { 4 } + 23 x ^ { 2 } - 50$
C) $f ( x ) = x ^ { 4 } - 46 x ^ { 2 } + 50$
D) $f ( x ) = x ^ { 4 } + 46 x ^ { 2 } + 50$

Question

Quizplus · Accepted Answer

Given the zeros $5i$ and $\sqrt{2}$, we must also include the conjugate of $5i$, which is $-5i$, to ensure the polynomial has real coefficients. The factors corresponding to these zeros are $x - 5i$, $x + 5i$, and $x - \sqrt{2}$. The polynomial is obtained by multiplying these factors and their conjugates. The factor for $5i$ and $-5i$ gives $x^2 + 25$, and the factor for $\sqrt{2}$ gives $x^2 - 2$. Multiplying these gives $x^4 + 23x^2 - 50$, which matches option B.

Write a Polynomial Function of Minimum Degree with Real Coefficients 5i5 i5i

Write a Polynomial Function of Minimum Degree with Real Coefficients $5 i$