Write a third-degree polynomial equation with real coefficients and the roots $8 - i , - 2$.
A) $\quad x ^ { 3 } + 16 x ^ { 2 } - 130 = 0$
B) none of these
C) $\quad x ^ { 3 } + 130 = 0$
D) $\quad x ^ { 3 } - 14 x ^ { 2 } + 33 x + 130 = 0$
E) $\quad x ^ { 3 } + 33 x ^ { 2 } + 16 x - 14 = 0$

Question

Quizplus · Accepted Answer

A polynomial with roots $8 - i, -2$ must have a factor of $x - (8-i) = x-8+i$ and a factor of $x-(-2)=x+2$. 
By conjugate pairs, we also know that the polynomial must have a factor of $x-(8+i)=x-8-i$. 
Multiplying these factors out, we get
$$(x+2)(x-8+i)(x-8-i) = (x+2)((x-8)+i)((x-8)-i)$$
$$= (x+2)(x^2-16x+65) = x^3-14x^2+33x+130$$
Thus, the answer is D.

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