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.

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 \( y = -x \).