Respuesta :
Answer:
[tex]x=0[/tex]
Step-by-step explanation:
We have been given that a polynomial function [tex]f(x)=3x^{5}-2x^{2}+7x[/tex] models the motion of a roller coaster. Every root of the polynomial represent when the roller coaster is at the ground level.
In order to find all the potential values of x when roller coaster is at the ground level, we find the zeroes of the polynomial by factoring it as shown below:
[tex]f(x)=3x^{5}-2x^{2}+7x\\f(x)=x(3x^{4}-2x+7)\\[/tex]
The zeroes of this polynomial occur when either [tex]x=0[/tex] or [tex]3x^{4}-2x+7=0[/tex].
The equation [tex]3x^{4}-2x+7=0[/tex] have no real solutions. Therefore, the only time when the roller coaster is at the ground is at [tex]x=0[/tex]
Answer:
At x=0 the roller coaster is at ground level. All the potential roots are
[tex]x=0,\pm 1, \pm 7, \pm \frac{1}{3}, \pm \frac{7}{3}[/tex].
Step-by-step explanation:
The given function is
[tex]f(x)=3x^5-2x^2+7x[/tex]
It is given that the roots of the function represent when the roller coaster is at ground level.
The factor form of given function is
[tex]f(x)=x(3x^4-2x+7)[/tex]
To find the roots of the function equate f(x)=0.
[tex]0=x(3x^4-2x+7)[/tex]
By using zero product property, equate each factor equal to 0.
[tex]x=0[/tex]
[tex]3x^4-2x+7=0[/tex] ...(1)
The equation (1) has no real root.
To find the potential root use rational root theorem.]
According to the rational root theorem, all the possible roots are in the form of
[tex]r=\pm \frac{\text{Factors of constant term}}{\text{Factors of leading term}} [/tex]
The leading term is 3 and the constant term is 7.
Factors of 7 are ±1, ±7 and the factors of 3 are ±1 and ±3.
All the possible roots are
[tex]x=\pm 1, \pm 7, \pm \frac{1}{3}, \pm \frac{7}{3}[/tex]
