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According to the Rational Root Theorem, what are all the potential rational roots of [tex]$f(x) = 5x^3 - 7x + 11$[/tex]?

A. [tex]\pm \frac{1}{11}, \pm \frac{5}{11}, \pm 1, \pm 5[/tex]
B. [tex]\pm \frac{1}{5}, \pm 1, \pm \frac{11}{5}, \pm 11[/tex]
C. [tex]\pm \frac{1}{11}, \pm \frac{1}{5}, \pm \frac{5}{11}, \pm 1, \pm \frac{11}{5}, \pm 5, \pm 11[/tex]
D. [tex]0, \pm \frac{1}{11}, \pm \frac{1}{5}, \pm \frac{5}{11}, \pm 1, \pm \frac{11}{5}, \pm 5, \pm 11[/tex]



Answer :

GinnyAnswer
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]

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