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

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- We first find the roots of the denominator
**x**^{4}- 5 x^{3}+ 3 x^{2}+ 15 x - 18 - Press 2nd POLY key.
- At the prompt
**order=**, type**4**and press the ENTER key. - At the prompt
**a4=**, type**1**and press the ENTER key. - At the prompt
**a3=**, type**-5**and press the ENTER key. - At the prompt
**a2=**, type**3**and press the ENTER key. - At the prompt
**a1=**, type**15**and press the ENTER key. - At the prompt
**a0=**, type**-18**and press the ENTER key. - Press F5 to pick
**SOLVE**. - The four roots of the numerator are displayed:
- We repeat the process above to find the roots of the denominator:
**x**^{6}- 15 x^{5}+ 78 x^{4}- 180 x^{3}+ 203 x^{2}- 165 x + 126 - Press the EXIT key followed by the 2nd POLY key.
- At the prompt
**order=**, type**6**and press the ENTER key. - At the prompt
**a6=**, type**1**and press the ENTER key. - At the prompt
**a5=**, type**-15**and press the ENTER key. - At the prompt
**a4=**, type**78**and press the ENTER key. - At the prompt
**a3=**, type**-180**and press the ENTER key. - At the prompt
**a2=**, type**203**and press the ENTER key. - At the prompt
**a1=**, type**-165**and press the ENTER key. - At the prompt
**a0=**, type**126**and press the ENTER key. - Press F5 to pick
**SOLVE**. - The six roots of the numerator are displayed:
- 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**. - 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. - The following is the graph of the denominator:
with the window

**[0, 8]**×**[-10, 10]**: - From the graph, it appears that
**3**may be a root of the denominator. We check this out to be so: - Hence, the roots of the denominator are:
**7, 3, 2, i, -i**where

**3**is repeated twice. - This tells us that
**x = 7**is vertical asymptote.- Since
**2**is a root of both the numerator and the denominator, the**2**is a removable discontinuity of**f**. - 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. **1.73205080757...**and**-1.73205080757...**, which are, in fact, the positive and negative square roots of**3**, are the only roots of**f**.- The following is the graph of
**f**:with the window

**[0, 8]**×**[-3, 3]**: