Question

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; ±1/3, ±1, ±7/3, ±7; 0, ±1/7, ±1, ±3/7, 3; 0, ±1/3, ±1, ±7/3, ±7

Explanation:

Step1: Factor out x

$f(x)=x(3x^{4}-2x + 7)$
Set $f(x) = 0$. Then either $x=0$ or $3x^{4}-2x + 7=0$. For the polynomial $a_nx^n+\cdots+a_1x + a_0$ (in our case of $3x^{4}-2x + 7$, $a_n = 3$, $a_0=7$), by the rational - root theorem, the possible rational roots are of the form $\pm\frac{p}{q}$, where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading - coefficient $a_n$. The factors of $a_0 = 7$ are $\pm1,\pm7$, and the factors of $a_n=3$ are $\pm1,\pm3$. So the possible rational roots of $3x^{4}-2x + 7$ are $\pm\frac{1}{3},\pm1,\pm\frac{7}{3},\pm7$.

Answer:

$0,\pm\frac{1}{3},\pm1,\pm\frac{7}{3},\pm7$