Questions: The polynomial function f(x)=3x^5-2x^2+7x models the motion of a roller coaster. The roots of the function represent when the roller coaster is at ground level. Which answer choice represents all potential values of when the roller coaster is at ground level? Begin by factoring x to create a constant term. ±1/7, ±1, ±3/7, ±3 0, ±1/3, ±1, ±7/3, ±7 ±1/3, ±1, ±7/3, ±7 0, ±1/7, ±1, ±3/7, 3

The polynomial function f(x)=3x^5-2x^2+7x models the motion of a roller coaster. The roots of the function represent when the roller coaster is at ground level. Which answer choice represents all potential values of when the roller coaster is at ground level? Begin by factoring x to create a constant term.

±1/7, ±1, ±3/7, ±3

0, ±1/3, ±1, ±7/3, ±7

±1/3, ±1, ±7/3, ±7

0, ±1/7, ±1, ±3/7, 3

Transcript text: The polynomial function $f(x)=3 x^{5}-2 x^{2}+7 x$ models the motion of a roller coaster. The roots of the function represent when the roller coaster is at ground level. Which answer choice represents all potential values of when the roller coaster is at ground level? Begin by factoring $x$ to create a constant term. \[ \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, \pm 3 \] \[ 0, \pm \frac{1}{3}, \pm 1, \pm \frac{7}{3}, \pm 7 \] \[ \pm \frac{1}{3}, \pm 1, \pm \frac{7}{3}, \pm 7 \] \[ 0, \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, 3 \]

Solution

Solution Steps

Step 1: Factor the Polynomial

The polynomial function given is

\[ f(x) = 3x^5 - 2x^2 + 7x. \]

By factoring out $ x $, we obtain:

\[ f(x) = x \left(3x^4 - 2x + 7\right). \]

Step 2: Identify the Roots

From the factorization, we can see that one root is

\[ x = 0. \]

Next, we need to find the roots of the quartic polynomial $ 3x^4 - 2x + 7 $. To determine the potential roots, we can apply the Rational Root Theorem, which suggests that any rational root, in the form $ \frac{p}{q} $, must have $ p $ as a factor of the constant term (7) and $ q $ as a factor of the leading coefficient (3).

The possible rational roots are:

\[ \pm 1, \pm 7, \pm \frac{1}{3}, \pm \frac{7}{3}. \]

Step 3: Evaluate the Quartic Polynomial

To find the actual roots, we can evaluate $ 3x^4 - 2x + 7 $ at the possible rational roots. However, since the polynomial does not have any real roots (as can be verified through further analysis or numerical methods), we conclude that the only root of the original polynomial is:

\[ x = 0. \]