To determine all the potential rational roots of the polynomial [tex]\( f(x) = 5x^3 - 7x + 11 \)[/tex] using the Rational Root Theorem, follow these steps:
1. Identify the constant term and the leading coefficient:
- The constant term of [tex]\( f(x) \)[/tex] is [tex]\( 11 \)[/tex].
- The leading coefficient (the coefficient of the highest degree term) is [tex]\( 5 \)[/tex].
2. List the factors of the constant term [tex]\( 11 \)[/tex]:
- The factors of [tex]\( 11 \)[/tex] are [tex]\( \pm 1 \)[/tex] and [tex]\( \pm 11 \)[/tex].
3. List the factors of the leading coefficient [tex]\( 5 \)[/tex]:
- The factors of [tex]\( 5 \)[/tex] are [tex]\( \pm 1 \)[/tex] and [tex]\( \pm 5 \)[/tex].
4. Form all possible combinations of the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient:
- Possible combinations are:
- [tex]\( \pm \frac{1}{1}, \pm \frac{1}{5}, \pm \frac{11}{1}, \pm \frac{11}{5} \)[/tex]
- Simplifying gives: [tex]\( \pm 1, \pm \frac{1}{5}, \pm 11, \pm \frac{11}{5} \)[/tex]
5. List out all these potential rational roots:
- [tex]\( \pm 1, \pm \frac{1}{5}, \pm 11, \pm \frac{11}{5} \)[/tex]
Hence, the potential rational roots of [tex]\( f(x) = 5x^3 - 7x + 11 \)[/tex] according to the Rational Root Theorem are:
[tex]\[ \pm 1, \pm \frac{1}{5}, \pm 11, \pm \frac{11}{5} \][/tex]
Thus, the correct option is:
[tex]\[
\pm \frac{1}{5}, \pm 1, \pm \frac{11}{5}, \pm 11
\][/tex]
