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The polynomial function [tex]f(x)=3x^5-2x^2+7x[/tex] 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 [tex]x[/tex] to create a constant term.

A. [tex]0, \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, \pm 3[/tex]

B. [tex]0, \pm \frac{1}{3}, \pm 1, \pm \frac{7}{3}, \pm 7[/tex]

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

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

Sagot :

GinnyAnswer
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].

First, we factor out [tex]\( x \)[/tex] from the polynomial:

[tex]\[ f(x) = x (3x^4 - 2x + 7) \][/tex]

Setting the equation to zero to find the roots:

[tex]\[ x (3x^4 - 2x + 7) = 0 \][/tex]

This gives us two parts to solve:

1. [tex]\( x = 0 \)[/tex]
2. [tex]\( 3x^4 - 2x + 7 = 0 \)[/tex]

The first root is straightforward:

[tex]\[ x = 0 \][/tex]

Next, we need to find the roots of the polynomial [tex]\( 3x^4 - 2x + 7 = 0 \)[/tex]. Solving quartic equations analytically can be complex, but we already know that the roots have been computed and presented in a detailed form. These roots are:

[tex]\[ \left[ \pm \sqrt{ \frac{14}{9 \left( \frac{1}{36} + \frac{\sqrt{5479}i}{108} \right)^{1/3}} + 2 \left( \frac{1}{36} + \frac{\sqrt{5479}i}{108} \right)^{1/3} }, \ \pm \sqrt{ -2 \left( \frac{1}{36} + \frac{\sqrt{5479}i}{108} \right)^{1/3} + \frac{4}{3 \sqrt{ \frac{14}{9 \left( \frac{1}{36} + \frac{\sqrt{5479}i}{108} \right)^{1/3}} + 2 \left( \frac{1}{36} + \frac{\sqrt{5479}i}{108} \right)^{1/3}}} - \frac{14}{9 \left( \frac{1}{36} + \frac{\sqrt{5479}i}{108} \right)^{1/3}} } \right] \][/tex]

Since the coefficients of the polynomial are real, the roots must also include complex conjugates, but the problem likely assumes if real roots are non-zero, they would be selected from the given options.

For the sake of this problem, let's check the answer choices for real-valued roots:
- Given that [tex]\( 0 \)[/tex] is a root, we seek any additional values in the options.
- A valid answer must then include [tex]\( 0 \)[/tex], inclusive available realistic values based on real quadratic solutions for factors of the complex-surrounded form stated as well.

From given options, only the first option includes 0:

```
0, \pm 1/7, \pm 1, \pm 3/7, \pm 3
```

That being justifiable arraying closer real-valued relations in correct amounts structures.

Thus, the correct answer for the roots of the function [tex]\( f(x) = 3x^5 - 2x^2 + 7x \)[/tex] representing the ground level values encompasses:

[tex]\[ 0, \pm \frac{1}{7}, \pm 1, \pm \frac{3}{7}, \pm 3 \][/tex]
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