A partial solution set is given for the equation. $$x ^ { 4 } - 9 x ^ { 3 } + 19 x ^ { 2 } - 9 x + 18 = 0 ; \quad \{ 3,6 \}$$
Find the complete solution set.
A) $\quad 3,6 , - i , i$
B) $3,3,3 , i$
C) none of these
D) $3,6,0,0$
E) $3,6 , - 6 i , 6 i$

Question

Quizplus · Accepted Answer

Since the given equation is a quartic equation (degree 4), we can try using polynomial division or synthetic division to find the quadratic factor to factorize the equation completely.
Using synthetic division, we get:
$$
\begin{array}{c|rrrrr}
&1 & - 9 & 19 & -9 & 18 \
\hline
3 & & - 6 & 39 & 0 & 18 \
6 & & 18 & 54 & 150 & 918
\end{array}
$$ 
So, the quadratic factor is $x^2 - 6x + 6$.
We can now solve it using the quadratic formula:
\begin{align*}
x &= \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(6)}}{2(1)} \
&= 3 \pm i\sqrt{3}.
\end{align*}
Thus, the complete solution set is $3, 6, 3 + i\sqrt{3}, 3 - i\sqrt{3}, 6 + i\sqrt{3}, 6 - i\sqrt{3}$. Simplifying, we get $3, 6, -i, i, 6\pm i\sqrt{3}$. The only option that includes all these roots is Choice A.

A Partial Solution Set Is Given for the Equation x4−9x3+19x2−9x+18=0;{3,6}x ^ { 4 } - 9 x ^ { 3 } + 19 x ^ { 2 } - 9 x + 18 = 0 ; \quad \{ 3,6 \}x4−9x3+19x2−9x+18=0;{3,6}

A Partial Solution Set Is Given for the Equation $x ^ { 4 } - 9 x ^ { 3 } + 19 x ^ { 2 } - 9 x + 18 = 0 ; \quad \{ 3,6 \}$