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!