Write a third-degree polynomial equation with real coefficients and the roots $6 , - i$.
A) none of these
B) $\quad x ^ { 3 } - 6 = 0$
C) $\quad x ^ { 3 } - 6 x ^ { 2 } + x - 6 = 0$
D) $\quad x ^ { 3 } - 7 x ^ { 2 } - x + 36 = 0$

Question

Quizplus · Accepted Answer

If the roots are 6 and -i, then the polynomial must also have a root of +i (since complex roots come in conjugate pairs). Thus the polynomial can be factored as:
$$(x-6)(x+i)(x-i) = (x-6)(x^2+1)$$
Expanding this gives:
$$x^3 - 6x^2 + x - 6 = 0$$
which matches option C.

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