Problem:
Using the TI-85 graphing calculator, determine the roots and the vertical asymptotes of the rational function

Visualization:

[Press here to see animation again!]

1. We first find the roots of the denominator

x4 - 5 x3 + 3 x2 + 15 x - 18

2. Press 2nd POLY key.
3. At the prompt order=, type 4 and press the ENTER key.
4. At the prompt a4=, type 1 and press the ENTER key.
5. At the prompt a3=, type -5 and press the ENTER key.
6. At the prompt a2=, type 3 and press the ENTER key.
7. At the prompt a1=, type 15 and press the ENTER key.
8. At the prompt a0=, type -18 and press the ENTER key.
9. Press F5 to pick SOLVE.
10. The four roots of the numerator are displayed:

11. We repeat the process above to find the roots of the denominator:

x6 - 15 x5 + 78 x4 - 180 x3 + 203 x2 - 165 x + 126

12. Press the EXIT key followed by the 2nd POLY key.
13. At the prompt order=, type 6 and press the ENTER key.
14. At the prompt a6=, type 1 and press the ENTER key.
15. At the prompt a5=, type -15 and press the ENTER key.
16. At the prompt a4=, type 78 and press the ENTER key.
17. At the prompt a3=, type -180 and press the ENTER key.
18. At the prompt a2=, type 203 and press the ENTER key.
19. At the prompt a1=, type -165 and press the ENTER key.
20. At the prompt a0=, type 126 and press the ENTER key.
21. Press F5 to pick SOLVE.
22. The six roots of the numerator are displayed:

23. Note that this time, the roots are represented as ordered pairs of real numbers. The reason for this is that some of the roots are complex numbers. The ordered pair (a, b) represents the complex number a + b i. For example, the root x5 is the number i and the root x6 is the number - i.
24. Also, the calculator indicates that the root x2 is 3 + 2.77778688887E-6. We want to investigate this further since the imaginary part of this number is very small.
25. The following is the graph of the denominator:

with the window [0, 8] × [-10, 10]:

26. From the graph, it appears that 3 may be a root of the denominator. We check this out to be so:

27. Hence, the roots of the denominator are:

7, 3, 2, i, -i

where 3 is repeated twice.

28. This tells us that

29. x = 7 is vertical asymptote.
30. Since 2 is a root of both the numerator and the denominator, the 2 is a removable discontinuity of f.
31. Since 3 is a root of both the numerator and the denominator but 3 is repeated twice in the denominator, x = 3 is also a vertical asymptote.
32. 1.73205080757... and -1.73205080757..., which are, in fact, the positive and negative square roots of 3, are the only roots of f.
33. The following is the graph of f:

with the window [0, 8] × [-3, 3]: