How do you identify all of the asymptotes of #y = (x^3 + 5x^2 + 3x + 10)/(x^2 + 2)#?

Question

Lily Anderson · Accepted Answer

y= x+5

Carry out the long division to write #y= x+5+(5x)/(x^2+2)# This shows that line y= x+5 is its asymptote.

Sadie Alexander · Answer

undefined To identify all asymptotes of $ y = \frac{x^3 + 5x^2 + 3x + 10}{x^2 + 2} $, follow these steps:

1. Vertical Asymptotes: Vertical asymptotes occur where the denominator of the rational function equals zero and the numerator does not. Set the denominator, $ x^2 + 2 $, equal to zero and solve for $ x $. Since $ x^2 + 2 $ is always positive, there are no real roots. Thus, there are no vertical asymptotes.

2. Horizontal Asymptotes: To find horizontal asymptotes, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $ y = 0 $. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In this case, the degree of the numerator is greater than the degree of the denominator. So, there is no horizontal asymptote.

3. Oblique (Slant) Asymptotes: If the degree of the numerator is exactly one greater than the degree of the denominator, there is an oblique asymptote. To find it, perform polynomial long division or use synthetic division to divide the numerator by the denominator. The quotient represents the equation of the oblique asymptote.

Since the degree of the numerator (3) is one more than the degree of the denominator (2), there is a slant asymptote. Perform polynomial long division or synthetic division to find it. After division, the quotient will be the equation of the oblique asymptote.

By following these steps, you can identify the asymptotes of the given rational function $ y = \frac{x^3 + 5x^2 + 3x + 10}{x^2 + 2} $.