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.
( 0,pm\frac{1}{3},pm1,pm\frac{7}{3},pm7 )
( 0,pm\frac{1}{7},pm1,pm\frac{3}{7},3 )
( pm\frac{1}{7},pm1,pm\frac{3}{7},pm3 )
( pm\frac{1}{3},pm1,pm\frac{7}{3},pm7 )

Explanation:

Step1: Factor out \(x\)

Given \(f(x)=3x^{5}-2x^{2}+7x\), factor out \(x\): \(f(x)=x(3x^{4}-2x + 7)\)
Set \(f(x) = 0\), then \(x = 0\) or \(3x^{4}-2x + 7=0\). For the polynomial \(P(x)=3x^{4}-2x + 7\), by the Rational Root Theorem, if \(P(x)\) has a rational root \(\frac{p}{q}\), \(p\) is a factor of the constant term \(7\) (so \(p=\pm1,\pm7\)) and \(q\) is a factor of the leading - coefficient \(3\) (so \(q = \pm1,\pm3\)). The possible rational roots are \(\pm\frac{1}{3},\pm1,\pm\frac{7}{3},\pm7\)

Answer:

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