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Polynomial long division is a method used to divide polynomials, much like how you might divide numbers. It's especially helpful in finding asymptotes in rational functions, where the degree of the numerator is greater than the degree of the denominator. To perform polynomial long division:- Arrange the terms of both the numerator and the denominator in descending order of the powers of their variables.- Divide the first term of the numerator by the first term of the denominator, and write this term as part of the quotient.- Multiply the entire denominator by this term and subtract the result from the numerator.- Bring down the next term from the numerator and repeat the process until no terms are left to bring down.In our exercise, dividing the cubic polynomial by the quadratic polynomial resulted in an expression of the form: \[ y = 2x + 5 \]This expression is the equation of the oblique asymptote, a topic we'll describe in more detail below.

Oblique asymptotes occur in rational functions when the degree of the numerator is exactly one more than the degree of the denominator. Instead of appearing as a horizontal or vertical line on the graph, these asymptotes appear as a slanted line that the graph approaches as the values of \(x\) get very large or very small.To find the oblique asymptote, polynomial long division is used, just like we did in our exercise. After dividing, the leading term in the quotient, before the remainder, becomes the equation of the asymptote.In the given function:- The numerator has a degree of 3.- The denominator has a degree of 2.Thus, an oblique asymptote is expected, and after division, we found it to be:\[ y = 2x + 5 \]This line indicates how the function behaves as \(x\) head towards infinity in both the positive and negative directions.

The domain and range of a rational function are crucial when graphing and analyzing it. They inform us about the scope and limitations of the function.- **Domain:** The domain of a rational function is all real numbers except where the denominator equals zero. However, in this case, when we attempted to find the roots of the denominator \(2x^2 - x + 6\), we found no real solutions, indicating no vertical asymptotes and hence no restrictions. Therefore, the domain is all real numbers.- **Range:** The range is established by observing both the graph and any asymptotic behavior. Since the oblique asymptote neither limits the function from reaching certain output values nor presents a horizontal line limiting values, the range spans all real numbers except for possible specific values such as y-intercepts if applicable.Thus, for our function, both the domain and range are all real numbers, indicating the function can take and produce any value without restriction.

Finding the intercepts of a rational function graph gives us crucial points that help in plotting the graph accurately and understanding its behavior.- **X-intercepts:** These are the points where the graph intersects the x-axis, ¡i.e., the output is zero. To find them, focus on the numerator: set \(4x^3 + 8x^2 - 36x - 72 = 0\) and solve for \(x\). These solutions are the x-coordinates of the intercepts since the entire function equals zero at these points.- **Y-intercept:** The y-intercept is found where the graph intersects the y-axis. This occurs when \(x\) is zero. Substituting \(x = 0\) into the function gives:\[ f(0) = \frac{-72}{6} = -12 \]Thus, the y-intercept is at the point \((0, -12)\).These intercepts provide key location references on the graph, enabling a clearer picture of the graph's interaction with axes and aiding in sketching its rough layout.