How do you find slant asymptotes of a rational function?

To find the slant (or oblique) asymptote of a rational function, you need to perform polynomial long division when the degree of the numerator is exactly one greater than the degree of the denominator.

Here’s a step-by-step guide:

1. Identify the Rational Function: Start with a rational function in the form f(x) = P(x) / Q(x), where P(x) is the numerator and Q(x) is the denominator.
2. Check the Degrees: Ensure that the degree of P(x) (numerator) is one more than the degree of Q(x) (denominator). If this condition is not met, there is no slant asymptote.
3. Perform Polynomial Long Division: Divide P(x) by Q(x). The quotient you get (disregarding the remainder) will be the equation of the slant asymptote.
4. Write the Equation: The result from the long division will give you a linear equation in the form of y = mx + b, which is the equation of the slant asymptote.

Example: Consider the function f(x) = (2x^2 + 3x + 1) / (x + 1). The degree of the numerator (2) is one greater than the degree of the denominator (1). Dividing 2x^2 + 3x + 1 by x + 1 gives us 2x + 1 as the quotient, which is the slant asymptote.

In summary, by ensuring the correct degree relationship and performing long division, you can successfully find the slant asymptotes of rational functions.