To solve the problem of finding a possible rational root for the polynomial [tex]\( F(x) = 3x^3 - x^2 + 4x + 5 \)[/tex] using the Rational Root Theorem, follow these steps:
1. Identify the constant term and the leading coefficient of the polynomial:
- The constant term is the term without any [tex]\( x \)[/tex], which is 5.
- The leading coefficient is the coefficient of the term with the highest power of [tex]\( x \)[/tex], which is 3.
2. List the possible rational roots:
According to the Rational Root Theorem, any rational root of the polynomial, expressed as [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, must be among the following:
- Factors of the constant term (5): [tex]\( \pm 1, \pm 5 \)[/tex]
- Factors of the leading coefficient (3): [tex]\( \pm 1, \pm 3 \)[/tex]
Hence, the possible rational roots are:
- [tex]\( \pm \frac{p}{q} \)[/tex] where [tex]\( p \)[/tex] is a factor of 5 and [tex]\( q \)[/tex] is a factor of 3.
This results in the possible roots being: [tex]\( \pm 1, \pm \frac{1}{3}, \pm 5, \pm \frac{5}{3} \)[/tex].
3. Match the given options to the possible roots:
Given options are:
- A. [tex]\(-7\)[/tex]
- B. [tex]\(\frac{4}{3}\)[/tex]
- C. [tex]\(-\frac{5}{3}\)[/tex]
- D. 6
4. Compare the options with the list of possible roots:
- [tex]\(-7\)[/tex] is not in the list of possible roots.
- [tex]\(\frac{4}{3}\)[/tex] is not in the list of possible roots.
- [tex]\(-\frac{5}{3}\)[/tex] is in the list of possible roots.
- 6 is not in the list of possible roots.
Upon comparison, we find that the only option from the given choices that matches one of the possible rational roots is:
C. [tex]\(-\frac{5}{3}\)[/tex]
Thus, the possible root of the polynomial [tex]\( F(x)=3x^3-x^2+4x+5 \)[/tex] among the given options is [tex]\( -\frac{5}{3} \)[/tex].
