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Long division in calculus is a technique to simplify the process of dividing polynomials. This method is similar to the standard long division you might remember from arithmetic. However, here it involves dividing a polynomial by another polynomial. For the exercise given, we need to divide the polynomial \(3x^2 - 1\) by \(x - 1\). The purpose is to find a simpler expression for the rational function, which will help us identify characteristics like slant asymptotes. When using long division on these polynomials:

• Start by dividing the highest degree term of the numerator by the highest degree term of the divisor. This gives you the first term of your quotient.
• Multiply the entire divisor by this term and subtract the result from the original polynomial.
• Repeat the process with the new polynomial obtained after subtraction, until the degree of the remainder is less than the degree of the divisor.

In our example, dividing \(3x^2\) by \(x\) results in \(3x\), the first term in our quotient. Continue this process until reaching the remainder, and you get \(3x + 3 - \frac{4}{x-1}\). The slant asymptote is found within the quotient, \(3x + 3\).

Polynomial division is a crucial skill when working with rational functions, especially to identify asymptotes. Understanding this concept will simplify and clarify how the function behaves. There are two main types of division you might consider:

• Long Division: As explained earlier, it involves a systematic approach where you divide one polynomial by another. It is very useful when dividing a higher degree polynomial by a lower one, as seen in the exercise above.
• Synthetic Division: This is a streamlined version of long division that you can utilize when dividing by a linear divisor of the form \(x - c\). It is quicker but only applicable under specific circumstances.

Returning to the exercise, using long division helps in determining the slant asymptote. When the degree of the polynomial in the numerator is exactly one more than that in the denominator, and there is no common factor, expect a slant (or oblique) asymptote in the form of the quotient without the remainder.

Graphing rational functions involves plotting the behavior and key characteristics of these functions, such as asymptotes. When working with functions like \(f(x) = \frac{3x^2 - 1}{x - 1}\), you must graph the function and the slant asymptote on the same set of axes. Start by identifying key features:

• X-intercepts: Points where the function crosses the x-axis, found by setting the numerator equal to zero.
• Vertical Asymptotes: These occur where the function is undefined, which in our function is at \(x=1\). This occurs when the denominator equals zero.
• Slant Asymptotes: As determined from long division, we previously found this to be \(y = 3x + 3\).

When graphing:

• Plot these asymptotes as dashed lines to represent tendencies where the function will draw near but not touch.
• Consider behavior at infinity to ensure the function approaches the slant asymptote.

Ultimately, the graph will indicate how \(f(x)\) behaves as \(x\) moves toward positive and negative infinity, showcasing that it approaches the slant asymptote, emphasizing the asymptotic nature of the expression.