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Rational functions are special types of mathematical expressions that involve ratios of two polynomials. These functions are capable of exhibiting a variety of interesting behaviors including asymptotes, holes, and intercepts. Understanding these characteristics is essential for graphing rational functions accurately.
The general form of a rational function is given as \[ f(x) = \frac{P(x)}{Q(x)} \]where \( P(x) \) and \( Q(x) \) are polynomials. It is important to note that \( Q(x) eq 0 \), as division by zero is undefined.
Rational functions can have different types of asymptotes: vertical, horizontal, or slant (oblique). Vertical asymptotes occur where the denominator equals zero and the rational function is undefined. Horizontal or slant asymptotes, indicating the end-behavior of the function, depend on the degrees of the numerator and denominator.
In our example, the rational function \[ f(x) = \frac{x^2 + x - 6}{x - 3} \]has a slant asymptote because the degree of the numerator is greater than the degree of the denominator by exactly one.

Polynomial long division is a technique used to divide one polynomial by another, similar to the method used in arithmetic for dividing numbers. It is particularly helpful in finding slant asymptotes for rational functions.
To perform polynomial long division, you need to follow a series of steps:

• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire divisor by the result and subtract it from the original polynomial.
• Repeat the process with the new polynomial that is formed after subtraction.

This process continues until the degree of the remainder is less than the degree of the divisor.
For our function \[ f(x) = \frac{x^2 + x - 6}{x - 3} \], we perform the division and find that \( x^2 + x - 6 \) divided by \( x - 3 \) yields a quotient of \( x + 4 \).
This quotient represents the equation of the slant asymptote, which is significant in understanding the function's graph.

The degree of a polynomial is a fundamental concept that influences the shape and properties of polynomial and rational functions. It is defined as the highest power of the variable in the polynomial expression.
For example, in the polynomial \( x^2 + x - 6 \), the degree is 2 because the highest power of \( x \) is 2. Similarly, for \( x - 3 \), the degree is 1.
The degree of the polynomials in a rational function helps determine the type of asymptotes present:

• If the degree of the numerator is greater than the degree of the denominator by exactly one, the function will have a slant asymptote.
• If the degrees are equal, the function will have a horizontal asymptote.
• If the degree of the denominator is greater than the numerator, the horizontal asymptote is at \( y = 0 \).

In our exercise, with \( f(x) = \frac{x^2 + x - 6}{x - 3} \), the degree of the numerator is 2, and the degree of the denominator is 1. This causes a slant asymptote in our function, which is precisely where the polynomial division becomes important.