- Verify that the degree of the numerator (3) is one higher than the degree of the denominator (2), confirming the existence of a slant asymptote.
- Divide \(10x^3 - 15x^2 + x - 1\) by \(5x^2 - 2\) using polynomial long division.
- Obtain the quotient \(2x - 3\) and identify it as the equation of the slant asymptote.
- Conclude that the slant asymptote is \(\boxed{y = 2x - 3}\), which corresponds to option a.
Explanation
- Analysis of the Rational Function
To find the slant (oblique) 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. Here, the numerator has degree 3 and the denominator has degree 2, confirming a slant asymptote exists. We write \(f(x)\) in the form \(mx + c + \frac{R(x)}{D(x)}\), where \(mx + c\) is the equation of the slant asymptote.
- First Step of 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 \(2x\) by the divisor \((5x^2 - 2)\):
\[2x(5x^2 - 2) = 10x^3 - 4x\]
Subtract this from the numerator to find the remainder:
\[(10x^3 - 15x^2 + x - 1) - (10x^3 - 4x) = -15x^2 + 5x - 1\]
- Second Step of Division
Now, divide the leading term of the new expression (\(-15x^2\)) by the leading term of the divisor (\(5x^2\)):
\[\frac{-15x^2}{5x^2} = -3\]
Multiply \(-3\) by the divisor \((5x^2 - 2)\):
\[-3(5x^2 - 2) = -15x^2 + 6\]
Subtract this from the current remainder to find the final remainder:
\[(-15x^2 + 5x - 1) - (-15x^2 + 6) = 5x - 7\]
Since the degree of the 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 rational 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 approaches the line \(y = 2x - 3\). Comparing this to the given choices, the correct slant asymptote is option (a).
Examples
In physics and engineering, slant asymptotes are frequently used to understand the long-term behavior of complex systems. For instance, if you are modeling the velocity of a rocket under certain drag conditions defined by a rational function, the slant asymptote can reveal the 'terminal slope' of the trajectory as time increases. Knowing this linear path allows engineers to simplify complex models and predict long-term trends without needing to compute the full function for every point in time.