Dividing polynomials using long division. So for this particular example, we are going to be dividing two polynomials except in this case our denominator is going to have a square terms, so it’s going to be of a degree 2.
How we divide this is like we divide anything else; any other polynomials or just big numbers. The first thing we want to do is make a division bracket and put our divisor on the outside. Now, this problem we also have no x term, okay? There’s no first degree term, and so in order to keep all my information straight, I’m going to throw it in with a coefficient of 0, because it’s not really there but that little 0 will help me keep my spot.
So my divisor then is 2x² plus 0x plus 1, okay. So then we’re going to write in our dividend underneath the square root, again making sure that every single term is represented. If you look over we don’t have a square term, so we want to throw in a 0 for our square term coefficient. So then we have 4x to the 4th, plus 6x³, plus 0x², plus 3x minus 1.
So now we want to divide this up and the first thing we want to do is get rid of this 4x to the 4th, okay? In order to do that we want to look at our leading term, our first term over here and figure out what we need to multiply this term by to get our 4x to the 4th. 2x² times 2x² is going to give us 4x to the 4th. And what I always do is I try to line up my powers so even though we’re dealing with this term over here, I’m going to keep my square terms together so all my squared terms are in a column, okay?
Remember when we multiply exponents, we add our, I’m sorry, when we multiply our bases we add our exponents. So 2x² times 2x² is actually 4x to the 4th. Multiplying this down the row, we end up with 0x³ and we have a 2x². We then subtract. 4x to the 4th minus 4x to the 4th, those cancel, 6x³ minus nothing is just going to leave us with 6x³ and 0x² minus 2x² is -2x². You could either drop this 3x down now or you can leave it up there just remembering it’s there, I prefer the later, it doesn’t matter.
So now we want to get rid of this 6x³. Look here in our leading term, we need to make 6x³ from that. If we multiply 2x² times 3x, we end up getting 6x³. So that goes up here, 3x, again multiplying through, 6x³, we’re not going to have an x² term because I wrote 0 and then +3x. Subtraction, x³ cancel, -2x² minus nothing is just going to be -2x². Remember we have this 3x up here, 3x minus -3x does go away so we have +0x-1.
Okay, now we want to get -2x² from 2x², we need to multiply by a -1. -1, -2x², our 0x stays there and -1 and subtract. Everything cancels out, everything is equal so we end up with 0. Tells us we have a remainder of 0 so this tells us that 2x² is actually a perfect divisor of this, leaving us with our answer 2x² plus 3x minus 1.
Well, long division, it’s a pretty involved process but it obeys the same rules as any other type of long division. Just write it out, when you’re missing something make sure you fill in the gaps to keep all your spaces clear then go through the process.