9. Which of the following is the equation of the oblique asymptote of the curve of equation $$ f(x)=\frac{2 x^{3}-13 x^{2}+17 x+12}{x^{2}-16} $$ ? A. $$ y=-2 x-13 $$ B. $$ y=2 x-13 $$ C. $$ y=2 x+13 $$ D. $$ y=-2 x+13 $$ E. None of the above

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We begin by noting that the degree of the numerator, $$2x^3 - 13x^2 + 17x + 12$$, is 3 and the degree of the denominator, $$x^2-16$$, is 2. Since the degree of the numerator is exactly one more than the degree of the denominator, there is an oblique (or slant) asymptote given by the quotient when the numerator is divided by the denominator.

**Step 1: Perform Polynomial Long Division**

Divide  
$$
2x^3 - 13x^2 + 17x + 12
$$
by  
$$
x^2 - 16.
$$

1. Divide the leading term $$2x^3$$ by $$x^2$$ to get $$2x$$.

2. Multiply the divisor by $$2x$$:
   $$
   2x \cdot (x^2 - 16) = 2x^3 - 32x.
   $$

3. Subtract this from the original numerator:
   $$
   (2x^3 - 13x^2 + 17x + 12) - (2x^3 - 32x) = -13x^2 + 49x + 12.
   $$

4. Next, divide $$-13x^2$$ by $$x^2$$ to get $$-13$$.

5. Multiply the divisor by $$-13$$:
   $$
   -13 \cdot (x^2 - 16) = -13x^2 + 208.
   $$

6. Subtract this from the current remainder:
   $$
   (-13x^2 + 49x + 12) - (-13x^2 + 208) = 49x + 12 - 208 = 49x - 196.
   $$

Thus, the result of the division is:
- **Quotient:** $$2x - 13$$
- **Remainder:** $$49x - 196$$

**Step 2: Identify the Oblique Asymptote**

The oblique asymptote is represented by the quotient, so we have:
$$
y = 2x - 13.
$$

**Conclusion**

The equation of the oblique asymptote is $$y = 2x - 13$$, which corresponds to option B.
The oblique asymptote is $$ y = 2x - 13 $$.

Which of the following is the equation of the oblique asymptote of the curve of equation
?
A.
B.
C.
D.
E. None of the above

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