What are the oblique and vertical asymptotes for the function y = (3x² - 5x + 2)/(x - 5)?

To identify the oblique and vertical asymptotes of the function ( y = \frac{3x^2 - 5x + 2}{x - 5} ), we first look for the vertical asymptotes. Vertical asymptotes occur when the denominator of a function is zero and the numerator is not zero at that same point. In this case, the denominator ( x - 5 ) is zero when ( x = 5 ). Therefore, the vertical asymptote is ( x = 5 ).

Next, we determine the oblique (or slant) asymptote. A rational function may have an oblique asymptote if the degree of the numerator is one more than the degree of the denominator. In this case, the degree of the numerator (which is ( 3x^2 - 5x + 2 )) is 2, and the degree of the denominator (which is ( x - 5 )) is 1. Since the degree of the numerator is exactly one greater than that of the denominator, we can find the oblique asymptote by performing polynomial long division of the numerator by the denominator.

Dividing ( 3x^2 -