Which of the following are roots of the polynomial function below?

Check all that apply.
[tex]F(x) = 2x^3 - 5x^2 + 2x + 1[/tex]

A. [tex]\frac{3+\sqrt{17}}{4}[/tex]
B. [tex]\frac{3-\sqrt{17}}{4}[/tex]
C. [tex]\frac{5-\sqrt{10}}{6}[/tex]
D. [tex]\frac{5+\sqrt{10}}{6}[/tex]
E. 1



Answer :

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]