To determine the values when the roller coaster is at ground level, we need to find the roots of the polynomial function [tex]\( f(x) = 3x^5 - 2x^2 + 7x \)[/tex]. These roots represent the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex].
First, let's factor out the common factor [tex]\( x \)[/tex] from the polynomial:
[tex]\[ f(x) = x (3x^4 - 2x + 7) \][/tex]
We now have the first root:
[tex]\[ x = 0 \][/tex]
Next, we need to solve for the roots of the polynomial equation [tex]\( 3x^4 - 2x + 7 = 0 \)[/tex]. Finding the exact roots of a quartic equation can be complex, but we can analyze the possible values given in the answer choices.
The choices provided are:
- [tex]\( 0, \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, \pm 3 \)[/tex]
- [tex]\( 0, \pm \frac{1}{3}, \pm 1, \pm \frac{7}{3}, \pm 7 \)[/tex]
- [tex]\( \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, \pm 3 \)[/tex]
- [tex]\( \pm \frac{1}{3}, \pm 1, \pm \frac{7}{3}, \pm 7 \)[/tex]
Since we factored out [tex]\( x \)[/tex] and found [tex]\( x = 0 \)[/tex] as a root, this immediately eliminates choices that do not include 0:
- [tex]\( \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, \pm 3 \)[/tex] (eliminated as it doesn’t include 0)
- [tex]\( \pm \frac{1}{3}, \pm 1, \pm \frac{7}{3}, \pm 7 \)[/tex] (eliminated as it doesn’t include 0)
This leaves us with:
- [tex]\( 0, \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, \pm 3 \)[/tex]
- [tex]\( 0, \pm \frac{1}{3}, \pm 1, \pm \frac{7}{3}, \pm 7 \)[/tex]
Let's confirm which set of roots can solve the quartic polynomial [tex]\( 3x^4 - 2x + 7 = 0 \)[/tex]. The coefficients of this polynomial suggest its roots are real and rational under specific conditions. Since the coefficients are straightforward, more likely possibilities for rational roots are simpler fractions and integers.
We would apply the rational root theorem to check for possible rational roots. This usually divides the constant term by the leading coefficient.
Given [tex]\(3x^4 - 2x + 7\)[/tex], neither [tex]\(\pm \frac{1}{7}, \pm \frac{3}{7}\)[/tex] nor [tex]\(\pm \frac{1}{3}, \pm \frac{7}{3}\)[/tex] look straightforward. After further inspection by checking with simpler factors [tex]\(\pm 1, \pm 3, 0\)[/tex], a polynomial of this form would have more feasible computation through simpler integer forms.
Through comparison and simplicity:
The suitable choice including simpler rational components possible to identify as not predicting complex roots confirms first format usually more coinciding polynomial analysis suggesting general integer/simpler fractions:
Thus the correct answer is:
[tex]\[ 0, \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, \pm 3 \][/tex]
Hence:
\[ \boxed{0, \pm\frac{1}{3}, \pm 1, \pm\frac{7}{3},\pm 7}
