- Analyze the rational function to identify that a slant asymptote exists because the numerator's degree is exactly one higher than the denominator's.
- Perform polynomial division to divide \(10x^3 - 15x^2 + x - 1\) by \(5x^2 - 2\).
- Determine the quotient of the division to be \(2x - 3\), which represents the linear path the function follows at its limits.
- Conclude that the equation of the slant asymptote is \(\boxed{y = 2x - 3}\).
Explanation
- Analysis of the Rational Function
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 division, dividing the numerator \(10x^3 - 15x^2 + x - 1\) by the denominator \(5x^2 - 2\). The quotient of this division will represent the equation of the slant asymptote \(y = mx + b\), as the rational function approaches this linear behavior as \(x\) becomes very large (positive or negative).
- Polynomial Long Division
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:
\[(10x^3 - 15x^2 + x - 1) - (10x^3 - 4x) = -15x^2 + 5x - 1\]
Now, divide the leading term of the new polynomial (\(-15x^2\)) by the leading term of the denominator (\(5x^2\)):
\[\frac{-15x^2}{5x^2} = -3\]
Multiply the divisor \(5x^2 - 2\) by \(-3\):
\[-3(5x^2 - 2) = -15x^2 + 6\]
Subtract this from our previous result:
\[(-15x^2 + 5x - 1) - (-15x^2 + 6) = 5x - 7\]
Since the degree of the remaining polynomial (\(5x - 7\)) is less than the degree of the denominator (\(5x^2 - 2\)), the division process is complete. The quotient is \(2x - 3\).
- Final Calculation and Conclusion
The quotient of the division is \(2x - 3\), and the remainder is \(5x - 7\). Thus, the original function can be written as:
\[f(x) = (2x - 3) + \frac{5x - 7}{5x^2 - 2}\]
As \(x \to \pm\infty\), the term \(\frac{5x - 7}{5x^2 - 2}\) approaches \(0\). Therefore, the function \(f(x)\) approaches the linear function \(y = 2x - 3\). Comparing this to the given options, the correct equation for the slant asymptote is \(y = 2x - 3\).
Examples
Slant asymptotes are essential tools in fields like data analysis and control systems engineering. When modeling a system whose output grows linearly over time but has small, decaying disturbances (represented by the remainder term), identifying the slant asymptote allows engineers to predict the long-term, stable behavior of the system. For instance, in an economics model where a company's profit trends toward a constant growth rate, the slant asymptote helps visualize the steady-state performance regardless of short-term fluctuations.