 In this video, we provide the solution to question number 10 for practice exam number three for math 1050, in which case we're given a rational function x to the fourth minus one over x squared minus four, and we're asked to compute the quotient and remainder for this rational function. That is to say, we need to do long division of polynomials. So let's give ourselves some of the space there. In the numerator, that's going to be our dividend here. We have x to the fourth, leave a space for x cubed, leave a space for x squared, leave a space for x, then we have a minus one right there. It's best to keep things well aligned. And then as our divisor, we have x squared minus four. Again, remember there's a spot for the x there, like so. And so then with this, we're going to get started. Take the leading term of the dividend, consider the leading term of the divisor. We're going to just quickly do the monomial division right there. x to the fourth divided by x squared is x squared. And so what we're going to do is we're going to take x squared and times it by the divisor x squared minus four. This gives us x to the fourth minus four x squared. We record this above, so x squared is part of the quotient. We times the divisor by x squared, which we did. This then gives us x to the fourth minus four x squared. We subtract this from above. So we see, of course, that the x to the fourths will cancel out. They should always cancel out. There's actually no x cubed terms whatsoever, so no big deal there. You have zero x squared minus minus four x squared, so that gives us a positive four x squared and then bring down the negative one, like so. Then we rinse and repeat. We take the leading coefficient of the new dividend, because this will shrink over time. And we compare that to the leading term of the divisor that doesn't change throughout the problem. And so we have this time four x squared divided by x squared. That gives us just a four. So that's what we're going to record on top. So we now have the quotient x squared plus four. But then we need to take the divisor and times everything by four, four times x squared minus four, like so. This will give us four x squared minus 16. We then record that here for x squared minus 16. We subtract this from above. Same thing again, the leading term should cancel out. Four x squared minus four x squared, they're gone. Then for the next one, we have a negative one minus negative 16. So 16 minus one is going to give us 15. This is going to be our remainder. You could record up here if you want to. X squared plus four remainder, 15. Honestly, the way I like to write it is in the following manner. So r of x, remember that was equal to x to the fourth minus one over x squared minus four. So this is then equal by the calculation we just did. This is x squared plus four and then plus 15 over x squared minus four. And so we should label this in some degree. This of course is our quotient and this is our remainder. And that then gives us the answer. The quotient is x squared plus four and the remainder is 15.