Determine if the function below has a horizontal or a slant asymptote, and explain why. $$ f(x)=\frac{-x^{3}-x+5 x^{2}+5}{-5 x+x^{2}+2} $$ Answer Attempt 1 out of 2 The function has a $$ \square $$ numerator is $$ \square $$ asymptote because the degree of the As $$ x $$ approaches infinity (positive or negative), the value of $$ y $$ approaches $$ \square $$ the degree of the denominator.

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To determine whether the function

$$
f(x) = \frac{-x^{3} + 5x^{2} - x + 5}{x^{2} - 5x + 2}
$$

has a horizontal or a slant (oblique) asymptote, we need to analyze the degrees of the numerator and the denominator.

### Step 1: Compare Degrees

- **Numerator:** The highest power of $$ x $$ is $$ x^3 $$, so the degree is **3**.
- **Denominator:** The highest power of $$ x $$ is $$ x^2 $$, so the degree is **2**.

### Step 2: Determine the Type of Asymptote

- **Horizontal Asymptote:** Exists if the degree of the numerator is **equal to or less than** the degree of the denominator.
- **Slant (Oblique) Asymptote:** Exists if the degree of the numerator is **exactly one more** than the degree of the denominator.
- **No Asymptote:** Exists if the degree of the numerator is more than one higher than the degree of the denominator.

In this case, the degree of the numerator (**3**) is **one more** than the degree of the denominator (**2**). Therefore, the function has a **slant asymptote**.

### Step 3: Find the Slant Asymptote

To find the slant asymptote, perform polynomial long division of the numerator by the denominator.

$$
\frac{-x^{3} + 5x^{2} - x + 5}{x^{2} - 5x + 2}
$$

**Division Process:**

1. **Divide the leading terms:** $$-x^3 \div x^2 = -x$$.
2. **Multiply the entire denominator by $$-x$$:**
   $$
   -x \times (x^{2} - 5x + 2) = -x^3 + 5x^2 - 2x
   $$
3. **Subtract this from the original numerator:**
   $$
   (-x^3 + 5x^2 - x + 5) - (-x^3 + 5x^2 - 2x) = x + 5
   $$
4. The remainder is $$x + 5$$, which has a lower degree than the denominator, so the division stops here.

**Result of Division:**
$$
f(x) = -x + \frac{x + 5}{x^{2} - 5x + 2}
$$

As $$ x $$ approaches infinity (both positive and negative), the term $$\frac{x + 5}{x^{2} - 5x + 2}$$ approaches **0**. Therefore, the slant asymptote is:

$$
y = -x
$$

### Conclusion

- **Type of Asymptote:** Slant (oblique) asymptote.
- **Equation of the Asymptote:** $$ y = -x $$.

### Final Answer

The function has a **slant asymptote** because the numerator has degree 3 and the denominator degree 2. As $$ x $$ approaches infinity or negative infinity, $$ y $$ approaches the line $$ y = -x $$.
The function has a slant asymptote $$ y = -x $$ because the degree of the numerator is one more than the degree of the denominator.

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