Dividing polynomials with remainders

we divide X to the third plus five X minus four by X minus X actually I think this is supposed to be an x squared let me correct it so we're going to divide X to the third plus five X minus four by x squared minus X plus one and just to see the different ways we can rewrite this we could rewrite this as X to the third plus 5x minus four divided by X to x squared minus X plus 1 or maybe the best way to write it in this circumstance since we're going to do algebraic long division is to write it as x squared minus X plus 1 divided into divide it into X to the third plus and actually I'm going to leave some blank space here we don't have an x degree at an x squared term here but I'm going to leave some space for it just so that we can align everything in the proper place when we actually do the division so X to the third plus nothing to the x squared power plus 5x minus 4 so we have a place for the third power the second power of the first power and the 0th power so now let's just do a little bit of algebraic long division let's look at the highest degree term x squared goes into X to the third how many times well it goes into it x times X to the third divided by x squared is equal to X to the 3 minus two which is equal to X to the 1 which is equal to X so it goes x times I'll write the X right over here and we multiply x times this entire thing x times x squared is X to the third x times negative x is negative x squared x times 1 is positive x and then we want to subtract this whole expression from that whole expression and that's the same thing as adding the opposite or multiplying each of these terms by negative 1 and then adding it to these terms so let's do that so we have negative x to the third negative 1 times negative x squared is positive x squared and then positive x times negative 1 is negative x and so let's now add everything X to the third minus Exeter those cancel out zero plus x squared gives us an x squared 5x minus X gives us a plus 4x and then we bring down this minus four we're not adding anything to it there you could view there's a zero here so let's bring down the minus four and now let's look at the highest degree terms x squared goes into x squared x squared goes into x squared exactly one time it's the same thing so we put a plus one and then you have one times x squared is x squared one times negative x is negative x one times 1 is 1 times 1 is 1 and now we want to subtract this from that or we want to add the opposite and to add the opposite we can just add multiply each of these terms by negative 1 x squared becomes negative x squared negative x times negative 1 is positive x and then positive 1 times negative 1 is negative 1 now let's do the addition x squared minus x squared they cancel out 4x plus X is 5x and then we have negative 4 minus 1 is minus is negative 5 now you might be tempted to keep dividing but you can't anymore this this term right here the highest degree term here is now higher than the highest degree term that you're going to try to divide into so we have a remainder we have a remainder we have a remainder so the answer to this is this expression right over here is equal to is equal to X plus 1 this X plus 1 plus the remainder plus 5x minus 5 whatever the remainder is divided by x squared x squared minus X plus 1 if this was divisible we could keep dividing but we're saying it's not it's now a lower degree than this down here so we could say it's X plus 1 X plus 1 plus whatever this remainder is divided by this thing over here so our answer I'm going to write it one more time it's X plus 1 plus 5x minus 5 over x squared minus X plus 1 and we can check that this works if we take this thing over here and we multiply it by this thing over here we should get the X to the third plus 5x minus 4 so let's do that let's multiply this thing let's multiply it by x squared minus X plus 1 and to do that let's just distribute this whole trinomial times each of these terms times each of these terms when we do the first term we have x squared minus 2x plus 1 times X so that's going to be x times x squared which is X to the third x times negative x which is negative x squared x times 1 which is plus X then we can multiply this whole thing times 1 so it's going to be plus x squared minus X plus 1 I'm just multiplying all of these times 1 and then we can multiply this whole thing times this thing now this is the same as the denominator here so it'll cancel out this will cancel with that and we're just going to be left we're just going to be left with the numerator over here so plus 5x minus 5 and now we can try to simplify it we only have one third-degree term the X to the 3rd so we have X to the third here second-degree terms we have a negative x squared and then we also have a positive x squared so they cancel out with each other first degree terms let's see we have a positive x and a negative x those cancel out with each other so we're just going to have that 5x over here so we're just going to have this 5x so then we have plus 5x and then we have the 0 the degree terms or the constant terms we have a positive 1 and a negative 5 add them together you get negative 4 so you get X to the 3rd plus 5x minus 4 which is exactly what we add over here