To determine the domain of the rational function [tex]\( R(x) = \frac{x}{x^3 - 216} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] that would make the denominator zero, as these values are where the function is undefined.
### Step-by-Step Solution
1. Identify the Denominator:
The denominator of [tex]\( R(x) \)[/tex] is [tex]\( x^3 - 216 \)[/tex].
2. Set the Denominator to Zero:
To find the values that make the denominator zero, we solve the equation:
[tex]\[
x^3 - 216 = 0
\][/tex]
3. Solve the Equation:
This can be solved by factoring or using the cubic root. Recognize that [tex]\( 216 = 6^3 \)[/tex], so:
[tex]\[
x^3 - 6^3 = 0
\][/tex]
Which gives us:
[tex]\[
x^3 = 216
\][/tex]
Taking the cubic root on both sides, we find:
[tex]\[
x = 6
\][/tex]
4. Identify Complex Roots:
In polynomial equations, there might be complex roots as well. Solving the polynomial [tex]\( x^3 - 216 \)[/tex] using factoring or specialized methods will give us:
[tex]\[
x = 6, \quad x = -3 - 3\sqrt{3}i, \quad x = -3 + 3\sqrt{3}i
\][/tex]
This means the denominator is zero at these [tex]\( x \)[/tex] values: [tex]\( 6 \)[/tex], [tex]\( -3 - 3\sqrt{3}i \)[/tex], and [tex]\( -3 + 3\sqrt{3}i \)[/tex].
5. Determine the Domain:
The domain of [tex]\( R(x) \)[/tex] must exclude the [tex]\( x \)[/tex] values that make the denominator zero. Hence, [tex]\( x \)[/tex] cannot be:
[tex]\[
6, \quad -3 - 3\sqrt{3}i, \quad -3 + 3\sqrt{3}i
\][/tex]
So, the domain of [tex]\( R(x) \)[/tex] is all real numbers excluding [tex]\( x = 6 \)[/tex]. The complex roots do not affect the domain for real number inputs, so we only have to exclude the real number root.
### Correct Option:
The correct choice given in the problem statement is:
A. The domain of [tex]\( R(x) \)[/tex] is [tex]\( \{ x \mid x \neq 6 \} \)[/tex].
You fill in the blank as [tex]\( \boxed{6} \)[/tex]. Hence, the completed answer choice is:
[tex]\[
\text{A. The domain of } R(x) \text{ is } \{ x \mid x \neq 6 \}.
\][/tex]
