Write a fourth-degree polynomial equation with real coefficients and the roots $i , 8 - i$.
A) $\quad x ^ { 4 } + 65 x ^ { 2 } - 16 x - 70 = 0$
B) $\quad x ^ { 4 } - 16 x ^ { 3 } - 65 x ^ { 2 } + 70 x + 16 = 0$
C) $\quad x ^ { 4 } + 66 x ^ { 2 } - 16 x + 65 = 0$
D) $\quad x ^ { 4 } - 16 x ^ { 3 } + 66 x ^ { 2 } - 16 x + 65 = 0$
E) none of these

Question

Quizplus · Accepted Answer

If a polynomial has real coefficients, its non-real roots must come in conjugate pairs. That is, if $a+bi$ is a root (where $a$ and $b$ are real numbers and $i$ is the imaginary unit), then $a-bi$ must also be a root. 
So, if the given roots are $i$ and $8-i$, then the other two roots must be their conjugates, which are $-i$ and $8+i$.
Thus, the polynomial can be written as: 
\begin{align*}
(x-i)(x+i)(x-(8-i))(x-(8+i)) &= ((x^2+1)(x-8+i))(x-8-i)\
&=(x^3-7x^2+17x-15)(x-8)\
&=\boxed{x^4-16x^3+66x^2-16x+65=0}
\end{align*}

Write a Fourth-Degree Polynomial Equation with Real Coefficients and the Roots