Question

according to the rational root theorem, what are all the potential rational roots of \\(f(x) = 5x^3 - 7x + 11\\)?

\\(\pm \frac{1}{11}, \pm \frac{5}{11}, \pm 1, \pm 5\\)

\\(\pm \frac{1}{5}, \pm 1, \pm \frac{11}{5}, \pm 11\\)

\\(\pm \frac{1}{11}, \pm \frac{1}{5}, \pm \frac{5}{11}, \pm 1, \pm \frac{11}{5}, \pm 5, \pm 11\\)

\\(0, \pm \frac{1}{11}, \pm \frac{1}{5}, \pm \frac{5}{11}, \pm 1, \pm \frac{11}{5}, \pm 5, \pm 11\\)

Explanation:

Identify the coefficients of the polynomial

We are given the polynomial function:

$$f(x) = 5x^3 - 7x + 11$$

The leading coefficient is \(a_n = 5\).
The constant term is \(a_0 = 11\).

Find the factors of the constant term and leading coefficient

According to the Rational Root Theorem, any potential rational root of the polynomial must be of the form \(\pm \frac{p}{q}\), where:

• \(p\) is a factor of the constant term \(a_0 = 11\).
• \(q\) is a factor of the leading coefficient \(a_n = 5\).

The factors of \(11\) are:

$$p \in \{1, 11\}$$

The factors of \(5\) are:

$$q \in \{1, 5\}$$

List all possible combinations of p/q

We form all possible fractions \(\pm \frac{p}{q}\) using the factors:

• For \(q = 1\):

$$\pm \frac{1}{1} = \pm 1$$

$$\pm \frac{11}{1} = \pm 11$$

• For \(q = 5\):

$$\pm \frac{1}{5}$$

$$\pm \frac{11}{5}$$

Combining these, the set of all potential rational roots is:

$$\pm \frac{1}{5}, \pm 1, \pm \frac{11}{5}, \pm 11$$

Match with the given options

Let's compare our list with the choices:

• Option 1: \(\pm \frac{1}{11}, \pm \frac{5}{11}, \pm 1, \pm 5\)
• Option 2: \(\pm \frac{1}{5}, \pm 1, \pm \frac{11}{5}, \pm 11\)
• Option 3: \(\pm \frac{1}{11}, \pm \frac{1}{5}, \pm \frac{5}{11}, \pm 1, \pm \frac{11}{5}, \pm 5, \pm 11\)
• Option 4: \(0, \pm \frac{1}{11}, \pm \frac{1}{5}, \pm \frac{5}{11}, \pm 1, \pm \frac{11}{5}, \pm 5, \pm 11\)

The correct list matches Option 2.

Answer:

• (A) \(\pm\frac{1}{11}, \pm\frac{5}{11}, \pm1, \pm5\)
• (B) \(\pm\frac{1}{5}, \pm1, \pm\frac{11}{5}, \pm11\) (Correct answer)
• (C) \(\pm\frac{1}{11}, \pm\frac{1}{5}, \pm\frac{5}{11}, \pm1, \pm\frac{11}{5}, \pm5, \pm11\)
• (D) \(0, \pm\frac{1}{11}, \pm\frac{1}{5}, \pm\frac{5}{11}, \pm1, \pm\frac{11}{5}, \pm5, \pm11\)