To determine the possible rational roots of the polynomial function [tex]\( F(x) = 2x^2 - 3x + 7 \)[/tex], we can use the Rational Root Theorem. According to the Rational Root Theorem, any rational root of a polynomial equation with integer coefficients is a fraction [tex]\(\frac{p}{q}\)[/tex], where:
- [tex]\( p \)[/tex] is a factor of the constant term (here, 7),
- [tex]\( q \)[/tex] is a factor of the leading coefficient (here, 2).
First, we identify the factors of the constant term (7) and the leading coefficient (2):
- The factors of 7 are [tex]\( \pm1 \)[/tex] and [tex]\( \pm7 \)[/tex].
- The factors of 2 are [tex]\( \pm1 \)[/tex] and [tex]\( \pm2 \)[/tex].
Next, we form all possible fractions [tex]\(\frac{p}{q}\)[/tex] using these factors. This yields the following combinations:
[tex]\[
\begin{align*}
\text{Possible rational roots} &= \{\pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{7}{1}, \pm \frac{7}{2}\} \\
&= \{\pm 1, \pm \frac{1}{2}, \pm 7, \pm 3.5\}
\end{align*}
\][/tex]
Now, we compare this list of possible rational roots with the given options:
- Option A: [tex]\( \pm 1 \)[/tex] - Yes, [tex]\( 1 \)[/tex] and [tex]\(-1\)[/tex] are on our list.
- Option B: [tex]\( \pm \frac{1}{7} \)[/tex] - No, [tex]\(\frac{1}{7}\)[/tex] and [tex]\(-\frac{1}{7}\)[/tex] are not on our list.
- Option C: [tex]\( \pm 7 \)[/tex] - Yes, [tex]\( 7 \)[/tex] and [tex]\(-7\)[/tex] are on our list.
- Option D: [tex]\( \pm 2 \)[/tex] - No, [tex]\( 2 \)[/tex] and [tex]\(-2\)[/tex] are not on our list.
- Option E: [tex]\( \pm \frac{1}{2} \)[/tex] - Yes, [tex]\(\frac{1}{2}\)[/tex] and [tex]\(-\frac{1}{2}\)[/tex] are on our list.
Therefore, the possible rational roots from the given options are:
- [tex]\(\pm 1\)[/tex]
- [tex]\(\pm 7\)[/tex]
- [tex]\(\pm \frac{1}{2}\)[/tex]
