To determine the vertical asymptotes of the function
[tex]\[ f(x) = \frac{x^5 + 4x^3 - 4x + 6}{x^3 - 2x^2 - 5x - 6}, \][/tex]
we need to find where the denominator equals zero, as these points cause the function to be undefined and thus might indicate vertical asymptotes.
### Step-by-Step Solution:
1. Identify the Denominator:
The denominator of the function is:
[tex]\[ x^3 - 2x^2 - 5x - 6 \][/tex]
2. Set the Denominator Equal to Zero:
We need to find the roots of the equation:
[tex]\[ x^3 - 2x^2 - 5x - 6 = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
Solve the cubic equation for [tex]\( x \)[/tex]:
[tex]\[ x = 2/3 + (-1/2 - \sqrt{3} i/2) (\sqrt{137}/3 + 134/27)^{1/3} + 19/(9(-1/2 - \sqrt{3} i/2)(\sqrt{137}/3 + 134/27)^{1/3}), \][/tex]
[tex]\[ x = 2/3 + (-1/2 + \sqrt{3} i/2) (\sqrt{137}/3 + 134/27)^{1/3} + 19/(9(-1/2 + \sqrt{3} i/2)(\sqrt{137}/3 + 134/27)^{1/3}), \][/tex]
[tex]\[ x = 2/3 + 19/(9 (\sqrt{137}/3 + 134/27)^{1/3}) + (\sqrt{137}/3 + 134/27)^{1/3} \][/tex]
These are the roots of the denominator. Because the roots are complex numbers, they do not correspond to vertical asymptotes in the real number plane.
### Conclusion:
Since all the roots of the denominator are complex, there are no vertical asymptotes for the function [tex]\( f(x) \)[/tex] in the real number plane.
