What is the domain of the rational function  f of x is equal to 2 x over the quantity 2 x cubed plus x squared minus 10 x end quantity  (2 points)

 x is an element of all real numbers such that x is not equal to 0

 x is an element of all real numbers such that x is not equal to 0 comma negative five halves comma 2

x is an element of all real numbers such that x is not equal to 0 comma five halves comma 2

x is an element of all real numbers such that x is not equal to negative five halves comma 2

To find the domain of the rational function \( f(x) = \frac{2x}{2x^3 + x^2 - 10x} \), we need to determine when the denominator equals zero, because the function is undefined where the denominator is zero.

### Steps:
1. The denominator is \( 2x^3 + x^2 - 10x \). We need to solve for \( x \) when this equals zero:
   \[
   2x^3 + x^2 - 10x = 0
   \]
2. Factor out \( x \):
   \[
   x(2x^2 + x - 10) = 0
   \]
   This gives us one solution: \( x = 0 \).

3. Now solve \( 2x^2 + x - 10 = 0 \). Using the quadratic formula:
   \[
   x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-10)}}{2(2)} = \frac{-1 \pm \sqrt{1 + 80}}{4} = \frac{-1 \pm \sqrt{81}}{4} = \frac{-1 \pm 9}{4}
   \]
   This gives two solutions:
   \[
   x = \frac{-1 + 9}{4} = \frac{8}{4} = 2 \quad \text{and} \quad x = \frac{-1 - 9}{4} = \frac{-10}{4} = -\frac{5}{2}
   \]

Thus, the function is undefined at \( x = 0 \), \( x = 2 \), and \( x = -\frac{5}{2} \).

### Conclusion:
The domain is all real numbers except \( x = 0 \), \( x = 2 \), and \( x = -\frac{5}{2} \).

The correct answer is:
**\( x \) is an element of all real numbers such that \( x \neq 0, -\frac{5}{2}, 2 \)**.

Would you like further details on factoring or solving quadratic equations?

### Follow-up Questions:
1. What happens to a rational function when the denominator equals zero?
2. How do you use the quadratic formula in general?
3. How can you verify if a solution makes a denominator zero?
4. What does the domain represent in the context of a function?
5. Why is it necessary to exclude certain values from the domain?

### Tip:
Always factor out common terms first when simplifying polynomials. This often makes solving easier!

What is the domain of the rational function f(x) = 2x / (2x^3 + x^2 - 10x)?

Learn how to find the domain of the rational function f(x) = 2x / (2x^3 + x^2 - 10x). This solution involves factoring polynomials and using the quadratic formula to determine the values of x where the function is undefined. Suitable for high school students in Grades 9-12.

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