- Identify the slant asymptote by performing polynomial long division on the given rational function \(f(x) = \frac{10x^3 - 15x^2 + x - 1}{5x^2 - 2}\).
- Divide the leading term \(10x^3\) by \(5x^2\) to obtain the first term of the quotient, \(2x\).
- Continue the division by subtracting \(-15x^2+6\) from the remainder, resulting in the constant term \(-3\) for the quotient.
- Conclude that the slant asymptote is represented by the linear quotient obtained, which is \(\boxed{y = 2x - 3}\).
Explanation
- Problem Analysis and Methodology
To find the slant asymptote of the rational function \(f(x) = \frac{10x^3 - 15x^2 + x - 1}{5x^2 - 2}\), we need to perform polynomial long division. A slant asymptote exists when the degree of the numerator is exactly one higher than the degree of the denominator, which is true here (degree 3 vs degree 2). We divide the numerator \(10x^3 - 15x^2 + x - 1\) by the denominator \(5x^2 - 2\).
- Performing Polynomial Division - Step 1
First, we divide the leading term of the numerator (\(10x^3\)) by the leading term of the denominator (\(5x^2\)):
\[\frac{10x^3}{5x^2} = 2x\]
Next, multiply the divisor \(5x^2 - 2\) by \(2x\):
\[2x(5x^2 - 2) = 10x^3 - 4x\]
Subtract this result from the original numerator:
\[(10x^3 - 15x^2 + x - 1) - (10x^3 - 4x) = -15x^2 + 5x - 1\]
- Performing Polynomial Division - Step 2
Now, we divide the leading term of the remainder (\(-15x^2\)) by the leading term of the divisor (\(5x^2\)):
\[\frac{-15x^2}{5x^2} = -3\]
Multiply the divisor \(5x^2 - 2\) by \(-3\):
\[-3(5x^2 - 2) = -15x^2 + 6\]
Subtract this from our previous remainder:
\[(-15x^2 + 5x - 1) - (-15x^2 + 6) = 5x - 7\]
Since the degree of the new remainder (\(5x - 7\), which is degree 1) is less than the degree of the divisor (\(5x^2 - 2\), which is degree 2), the division process is complete. The quotient is \(2x - 3\).
- Identifying the Slant Asymptote
The result of the division can be written as:
\[f(x) = 2x - 3 + \frac{5x - 7}{5x^2 - 2}\]
As \(x\) approaches positive or negative infinity, the fraction \(\frac{5x - 7}{5x^2 - 2}\) approaches \(0\). Therefore, the function \(f(x)\) behaves like the linear function \(y = 2x - 3\) as \(x\) grows large. This linear expression is the slant asymptote of the function.
- Final Answer
The equation of the slant asymptote of the given rational function is \(\boxed{y = 2x - 3}\).
Examples
Slant asymptotes are essential for understanding the long-term behavior of rational functions, which frequently appear in modeling growth and decay processes. For example, if a cost-per-unit function for manufacturing reaches a steady growth rate, the slant asymptote helps predict the long-term trend of the costs. Identifying this linear guide allows businesses to plan for future expenses by recognizing the linear behavior the function approaches as production volume increases.