To determine which of the given values are potential roots of the polynomial function [tex]\( f(x) = 3x^3 - 13x^2 - 3x + 45 \)[/tex], we need to evaluate the polynomial at each value and check if the result is zero.
Let's evaluate [tex]\( f(x) \)[/tex] for each given value:
1. [tex]\( f\left(\frac{1}{3}\right) \)[/tex]
2. [tex]\( f\left(-\frac{1}{3}\right) \)[/tex]
3. [tex]\( f(5) \)[/tex]
4. [tex]\( f(-5) \)[/tex]
5. [tex]\( f\left(\frac{5}{3}\right) \)[/tex]
6. [tex]\( f\left(-\frac{5}{3}\right) \)[/tex]
7. [tex]\( f(9) \)[/tex]
8. [tex]\( f(-9) \)[/tex]
9. [tex]\( f(1) \)[/tex]
10. [tex]\( f(-1) \)[/tex]
11. [tex]\( f(15) \)[/tex]
12. [tex]\( f(-15) \)[/tex]
13. [tex]\( f(3) \)[/tex]
14. [tex]\( f(-3) \)[/tex]
15. [tex]\( f(45) \)[/tex]
16. [tex]\( f(-45) \)[/tex]
Out of all these possible values, we look for the one where [tex]\( f(x) = 0 \)[/tex]. When [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 3(3)^3 - 13(3)^2 - 3(3) + 45
\][/tex]
[tex]\[
= 3(27) - 13(9) - 3(3) + 45
\][/tex]
[tex]\[
= 81 - 117 - 9 + 45
\][/tex]
[tex]\[
= 0
\][/tex]
Therefore, the correct value among the given options that is a potential root of [tex]\( f(x) = 3x^3 - 13x^2 - 3x + 45 \)[/tex] is [tex]\( 3 \)[/tex]. Thus, the only potential root from the given values is [tex]\( 3 \)[/tex].
