 In this video, we provide the solution to question number five from practice exam number three for math 1050 We're given a rational function g of x equals 3x to the fourth minus x squared over x cubed minus x squared plus one and we're asked to find any Oblique asymptotes if there's any well, of course if there's any there's only one And an oblique asymptote exists exactly when the numerator has a degree larger than the denominator by one So since it's four over three the oblique asymptote does exist And that oblique asymptote is going to be the quotient when we take the numerator divided by the denominator We don't need the remainder Since the denominator of course is a cubic polynomial. We're gonna have to use some long division here But it's not as long as it might sound We have 3x to the fourth. There's no x cubed term So I'm gonna skip that x squared and then no other term So just kind of consider it there and then the then the divisor is x cubed Minus x squared plus one. There's no linear term. That's okay So we ask ourselves how many times this x cubed divide into 3x to the fourth? Essentially we're asking ourselves 3x to the fourth divided by x cubed. That's going to give us 3x So we then record that over here like so Excuse me 3x So then we're going to take the divisor and times it by 3x 3x times 3x cubed is 3x to the fourth that should match up Then we're gonna take 3x times negative x squared, which gives you a negative 3x cubed We have to subtract this from above That 3x to the fourth will cancel out. You'll take zero x cubed minus minus 3x cubed So that's gonna give you a positive 3x cubed bring down the negative x squared and we can keep on going So then we ask the same question. How many times is x cubed go into 3x cubed? Well, again, that's just 3x cubed divided by x cubed. That's gonna give us a 3 So we put in we put in there the plus 3 and now at this moment We can actually stop going because we we're doing long division, but we don't have to do the entire thing We're just looking for the quotient. So at this moment Every other calculation we do would be going towards the remainder of this division Which we don't need the oblique asymptote is the quotient and therefore we see the correct answer is D The oblique asymptote is the line y equals 3x plus 3