 In this video, we provide the solution to question number 16 from practice the practice final exam for math 1050 We have to compute the remainder in quotient of the rational function r of x equals 3x to the fourth minus x cubed plus x minus 2 all over 3x squared plus x plus 1 as the denominator is a quadratic Function we get we can't use synthetic division to do division here. We're gonna have to use long division So let's set that thing up. So our divisor is 3x squared plus x plus 1 We need to then divide the numerator 3x to the fourth Minus x cubed. There's no x squared terms. I actually gonna leave a space for it Plus x minus 2 like so So then we ask ourselves how many times is 3x squared divided into 3x to the fourth That's gonna happen exactly x squared times So that's why I left a gap for the x squared term there So we get what we're doing here is 3x to the fourth divided by 3x squared That equals just x squared. That's where this number came from So next what we have to do is we have to take x squared and times it by our divisor 3x squared plus x plus 1 Distribute this throughout and you end up with 3x to the fourth plus x cubed plus x squared like so I'm gonna record that over here. So we have a 3x to the fourth Plus x cubed plus x squared and then we subtract this from above The the first term should always cancel out if we chose this number correctly Then be careful with this because you have a double negative. Why I shouldn't say you have a double negative You have negative here then this distributes. It's such a single negative JK on that one But you're gonna get a negative x cubed minus x cubed So the thing is they actually combine to give you a two negative 2x cubed and then as you have no x squared You have 0x squared minus x squared which gives you a negative x squared like so Bring down the next term. You know you could bring down all the terms We actually won't need the negative 2 yet. So I'm not gonna bring it down yet Then we repeat this process How many times this 3x squared divide into negative 2x cubed and that situation we'd end up with a negative 2 thirds X like so so again, what are we doing here? We're just computing the ratio What is computing the ratio negative 2x cubed divided by 3x squared that simplifies to be negative 2 Thirds x like so. That's the number that went up here Then we have to take This partial quotient two-thirds x and times it by the divisor 3x squared plus x plus 1 So with the first one you end up with a negative 2x cubed for the next one You're gonna negative two-thirds x squared and another negative two-thirds x like so We then record that over here negative 2x cubed minus two-thirds x squared minus two-thirds x like so We then subtract this from above The first term should cancel out perfectly the next one's maybe not so much we do have some fractions here So be careful here. So you're gonna subtract 1x squared and then this is a double negative So you have you're gonna take one Just it should say sorry negative one plus two-thirds Negative one of course we can write that as negative three-thirds plus two-thirds So that's gonna give us a negative one-third when we're done That's the coefficient of x squared that we're gonna record right here negative one-third x squared Similar thing for the next one you will distribute this negative sign here So that's gonna be a one plus two-thirds one plus two-thirds this time. So again, it's three-thirds plus two-thirds That's gonna give us five-thirds. That's the coefficient of x that we need to put in there So we get five-thirds x like so bring down the negative two and then we have to do this one more time So again, we have the leading term here negative one-third x squared We divide that by three that actually is going to give us a negative one-ninth Right there as the quotient. So at this moment actually I'm gonna put a label here. We found our quotient Let's actually label this this is our quotient Because we're asked to identify the quotient. So we have the quotient, but we need to find the remainder So we're not quite done with that yet What we have to do is we have to take this thing and times it by negative one-ninth So three x squared times negative one-ninth will be negative one-third x squared X times negative one-ninth will be one negative one-ninth x and then negative one-ninth times one is negative one-ninth like so We subtract this from above Again, the leading term should cancel out perfectly For the next one might take a little bit more effort here because we have some fractions to deal with So we have five-thirds plus one-ninth So we should rewrite the five-thirds as 15 ninths plus one ninth. That's gonna give us 16 ninths there So we record that 16 over 9 x and then for the last part there we have a negative two plus one ninth Negative two plus one ninth. So again, I'm gonna rewrite the denominator. So it's a nine there So I'm gonna get negative 18 over 9 plus one ninth like so And so that'll then combine to give us a negative 17 ninths in which case that then gives us The rest of the remainder negative 17 ninths like so Maybe put a box or something to indicate that we've now found the remainder here Clearly indicate what we have and so zooming out We could then see the quotient was x squared minus two-thirds x minus one ninth And then the remainder was 16 16 ninths x minus 17 ninths