Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the oblique asymptote of \(g(x)=\frac{4 x^{3}+6 x^{2}-3 x+1}{2 x^{2}-4 x+3}\).

Short answer

The oblique asymptote of the function is y = 2x + 7.

Step-by-step solution

- Identify Degree of Numerator and Denominator

Check the degrees of the numerator and denominator polynomials. The numerator is of degree 3 and the denominator is of degree 2.

- Determine Presence of Oblique Asymptote

Since the degree of the numerator (3) is exactly one more than the degree of the denominator (2), there is an oblique asymptote.

- Perform Polynomial Long Division

Divide the numerator by the denominator using polynomial long division. Divide the leading term of the numerator, 4x^3, by the leading term of the denominator, 2x^2, to get 2x. Multiply the entire denominator by 2x and subtract from the numerator. Repeat the process for the remaining terms.

- Simplify the Result

Keep dividing until reaching a remainder with degree less than the denominator's degree. The quotient without the remainder represents the equation of the oblique asymptote.

- Write the Oblique Asymptote

The result of the polynomial long division (without the remainder) gives the oblique asymptote. This will be a linear equation in the form of y = mx + b.

Key concepts

Rational Functions

Rational functions are fractions where the numerator and the denominator are both polynomials. In an equation like this, you might see something like \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are the polynomials.
These functions can have different types of asymptotes, which are lines that the graph approaches but never touches.
Asymptotes come in three forms: vertical, horizontal, and oblique. Which ones you get depend on the degrees of the polynomials in the numerator and the denominator.

Polynomial Long Division

To find an oblique asymptote, you need to use polynomial long division. This is a process where you divide one polynomial by another, just like you do with regular numbers.
Here's the basic idea:

• Take the leading term of the numerator and divide it by the leading term of the denominator.
• Multiply the entire denominator by this result and subtract it from the numerator.
• Repeat this process with the new numerator (the remainder) until you can't divide anymore (because the degree of the remainder is less than that of the denominator).

For the given problem, divide the term \(4x^3 \) by \(2x^2 \) to get \(2x \). Then subtract the product of \(2x \) and the denominator from the numerator. Keep repeating this process with the new terms until you're left with a remainder.

Asymptotes in Algebra

Asymptotes in algebra are important because they give you an idea of the behavior of a graph at its extremes. There are three main types of asymptotes:

• Vertical Asymptotes: occur where the denominator equals zero (and the numerator does not equal zero at the same point). The graph approaches these lines but never touches or crosses them.
• Horizontal Asymptotes: found when the degrees of the numerator and denominator are the same, or when the degree of the denominator is higher than the numerator. This gives a line that the graph flattens out to at extreme values of x.
• Oblique Asymptotes: happen when the degree of the numerator is exactly one higher than the degree of the denominator. This results in a slanted line.

In this problem, \( g(x) = \frac{4x^3 + 6x^2 - 3x + 1}{2x^2 - 4x + 3} \), you get an oblique asymptote because the numerator's degree (3) is one more than the denominator's degree (2). The equation of this oblique asymptote will be the result of the polynomial long division, excluding the remainder.