To determine which values are roots of the polynomial [tex]\( F(x) = 2x^3 - x^2 - 9x + 6 \)[/tex], we need to evaluate the polynomial at each given value. A value [tex]\(a\)[/tex] is a root of the polynomial if [tex]\( F(a) = 0 \)[/tex].
Let's evaluate the polynomial [tex]\(F(x)\)[/tex] at each given value:
### Evaluation at [tex]\( x = \frac{9+\sqrt{55}}{4} \)[/tex]
[tex]\[ F\left(\frac{9+\sqrt{55}}{4}\right) = 2\left(\frac{9+\sqrt{55}}{4}\right)^3 - \left(\frac{9+\sqrt{55}}{4}\right)^2 - 9\left(\frac{9+\sqrt{55}}{4}\right) + 6 \][/tex]
After evaluating, we find that this does not simplify to zero. Therefore,
[tex]\[ \frac{9+\sqrt{55}}{4} \text{ is not a root.} \][/tex]
### Evaluation at [tex]\( x = 2 \)[/tex]
[tex]\[ F(2) = 2(2)^3 - (2)^2 - 9(2) + 6 \][/tex]
[tex]\[ F(2) = 2(8) - 4 - 18 + 6 \][/tex]
[tex]\[ F(2) = 16 - 4 - 18 + 6 \][/tex]
[tex]\[ F(2) = 0 \][/tex]
Thus,
[tex]\[ 2 \text{ is a root.} \][/tex]
### Evaluation at [tex]\( x = \frac{9-\sqrt{55}}{4} \)[/tex]
[tex]\[ F\left(\frac{9-\sqrt{55}}{4}\right) = 2\left(\frac{9-\sqrt{55}}{4}\right)^3 - \left(\frac{9-\sqrt{55}}{4}\right)^2 - 9\left(\frac{9-\sqrt{55}}{4}\right) + 6 \][/tex]
After evaluating, we find that this does not simplify to zero. Therefore,
[tex]\[ \frac{9-\sqrt{55}}{4} \text{ is not a root.} \][/tex]
### Evaluation at [tex]\( x = \frac{-3+\sqrt{33}}{4} \)[/tex]
[tex]\[ F\left(\frac{-3+\sqrt{33}}{4}\right) = 2\left(\frac{-3+\sqrt{33}}{4}\right)^3 - \left(\frac{-3+\sqrt{33}}{4}\right)^2 - 9\left(\frac{-3+\sqrt{33}}{4}\right) + 6 \][/tex]
After evaluating, we find that this simplifies to zero. Therefore,
[tex]\[ \frac{-3+\sqrt{33}}{4} \text{ is a root.} \][/tex]
### Evaluation at [tex]\( x = \frac{-3-\sqrt{33}}{4} \)[/tex]
[tex]\[ F\left(\frac{-3-\sqrt{33}}{4}\right) = 2\left(\frac{-3-\sqrt{33}}{4}\right)^3 - \left(\frac{-3-\sqrt{33}}{4}\right)^2 - 9\left(\frac{-3-\sqrt{33}}{4}\right) + 6 \][/tex]
After evaluating, we find that this simplifies to zero. Therefore,
[tex]\[ \frac{-3-\sqrt{33}}{4} \text{ is a root.} \][/tex]
So, the roots of the polynomial [tex]\(F(x) = 2x^3 - x^2 - 9x + 6\)[/tex] corresponding to the given options are:
B. 2
D. [tex]\(\frac{-3+\sqrt{33}}{4}\)[/tex]
E. [tex]\(\frac{-3-\sqrt{33}}{4}\)[/tex]
