To determine which of the given values are roots of the polynomial function [tex]\( F(x) = 2x^3 - 5x^2 + 2x + 1 \)[/tex], we need to check if substituting each value into the polynomial results in zero.
### Step-by-Step Solution:
1. Evaluate [tex]\( F \left( \frac{3 + \sqrt{17}}{4} \right) \)[/tex]:
[tex]\[
F \left( \frac{3 + \sqrt{17}}{4} \right) \neq 0
\][/tex]
This value is not a root of [tex]\( F(x) \)[/tex].
2. Evaluate [tex]\( F \left( \frac{3 - \sqrt{17}}{4} \right) \)[/tex]:
[tex]\[
F \left( \frac{3 - \sqrt{17}}{4} \right) = 0
\][/tex]
This value is a root of [tex]\( F(x) \)[/tex].
3. Evaluate [tex]\( F \left( \frac{5 - \sqrt{10}}{6} \right) \)[/tex]:
[tex]\[
F \left( \frac{5 - \sqrt{10}}{6} \right) \neq 0
\][/tex]
This value is not a root of [tex]\( F(x) \)[/tex].
4. Evaluate [tex]\( F \left( \frac{5 + \sqrt{10}}{6} \right) \)[/tex]:
[tex]\[
F \left( \frac{5 + \sqrt{10}}{6} \right) \neq 0
\][/tex]
This value is not a root of [tex]\( F(x) \)[/tex].
5. Evaluate [tex]\( F(1) \)[/tex]:
[tex]\[
F(1) = 0
\][/tex]
This value is a root of [tex]\( F(x) \)[/tex].
### Conclusion:
The roots of the polynomial [tex]\( F(x) = 2x^3 - 5x^2 + 2x + 1 \)[/tex] from the given options are:
- [tex]\( \frac{3 - \sqrt{17}}{4} \)[/tex] (Option B)
- [tex]\( 1 \)[/tex] (Option E)
Therefore, the correct answers are:
[tex]\[
\boxed{B \text{ and } E}
\][/tex]
