In simplifying algebraic fractions, we occasionally need a process known as. Polynomial. Division. Before we do that, I want to take you back to something that you actually know very well indeed, and that's ordinary long division. You know how to do long division, but I want to go over it again. 'cause I want to point out certain things to you. 'cause the things that are important about long division are also important in polynomial division so. Let's have a look at a long division. Some supposing I want to divide 25. Into Let's say 2675. When I would have to do is look at 25 in tool 2. No way 25 into 26. It goes once and write the one there. Add multiply the one by the 25. And subtract and have one left. Hope you remember doing that. You were probably taught how to do that at primary school or the beginnings of Secondary School. Next step is to bring down the next number, so we bring down 17. Well, we bring down Seven to make it 17 and now we say how many times does 25 going to 17. It doesn't go at all. It's not enough, so we have to record the fact that it doesn't go with a 0. Next we bring down the five. So now we've got 175 and we say how many times does 25 go into that? And it goes 7 and we can check that Seven 535, five down three to carry. 7 twos are 14 and three is 17. Tracked, we get nothing left. So this is our answer. We've nothing left there, no remainder, nothing left over. And there's our answer. 2675 divides by 25 and the answer is 107. They just look at what we did. We did 25 into 26 because that went. We then recorded that once that it went there, multiplied, wrote the answer and subtracted. We brought down the next number. Asked how many times 25 went into it, it didn't go. We recorded that and brought down the next number. Then we said how many times does 25 going to that Seven we did the multiplication, wrote it down, subtracted, got nothing left so it finished. What we're going to do now is take that self same process and do it with algebra. So let us take. This 27 X cubed. +9 X squared. Minus 3X. Minus 10. All over. 3X minus 2. We want to divide that into that. We want to know how many times that will fit into there, so we set it up exactly like a long division. Problem by dividing by this. This is what we're dividing into 27 X cubed plus nine X squared minus three X minus 10. So we ask ourselves, how many times does well? How many times does that go into that? But difficult what we ask ourselves is how many times does the excpet go into this bit? Just like we asked ourselves how many times the 25 went into the 26, how many times does 3X? Go into 27 X cubed. The answer must be 9 X squared because Nynex squared times by three X gives us 27 X cubed and we need to record that. But we need to record it in the right place and because these are the X squared's we record that above the X squares. So now we do the multiplication. Nine X squared times 3X is 27 X cubed. Nine X squared times minus two is minus 18 X squared. Just like we did for long division, we now do the Subtraction. 27 X cubed takeaway 27 X cubed none of them, because we arrange for it to be so Nynex squared takeaway minus 18 X squared gives us plus 27 X squared. Now we do what we did before we bring down the next one, so we bring down the minus 3X. How many times does 3X go into 27 X squared? Answer. It goes 9X times and we write that in the X Column. So now we have 9X times 3 X 27 X squared 9X times, Y minus 2 - 18 X. And we subtract again. 27 X squared takeaway, 27 X squared, no X squared, but we arrange for it to be like that, minus three X minus minus 18X. Well, that's going to give us plus 15X altogether, and we bring down the minus 10. 3X into 15X. This time it goes five times, so we can say plus five there. And again it's in the numbers. The constants column at the end. Five times by 15 times by three X gives us 15X. Write it down there five times by minus two gives us minus 10 and we can see that when we take these two away. Got exactly the same expression. 15X minus 10 takeaway. 50X minus 10 nothing left. So there's our answer, just as in the long division. The answer was there. It's there now so we can say that this expression is equal to 9 X squared plus 9X. Plus 5. Let's take another one. So we'll take X to the 4th. Plus X cubed. Plus Seven X squared minus six X +8. Divided by all over X squared, +2 X +8. So this is what we're dividing by and this is what we're dividing into is not immediately obvious what the answer is going to be. Let's have a look X squared plus 2X plus 8IN tool. All of this. Our first question is how many times does X squared going to X to the 4th? We don't need to worry about the rest, we just do it on the first 2 bits in each one, just as the same as we did with the previous example. How many times X squared going to X to the four will it goes X squared times? So we write it there over the X squared's. Now we do the multiplication X squared times. My X squared is X to the 4th. X squared by two X is plus 2X cubed X squared by 8 is plus 8X squared. And now we do the Subtraction X. The four takeaway X to the 4th there Arnold, but we arranged it that way. X cubed takeaway 2X cubed minus X cubed. Seven X squared takeaway, 8X squared minus X squared and bring down the next term. Now we say how many times does X squared going to minus X cubed, and it must be minus X, and so we write it in the X Column. And above the line there, next the multiplication minus X times by X squared is minus X cubed minus X times 2X is minus two X squared and minus X times by 8 is minus 8X. Do the subtraction minus X cubed takeaway minus X cubed. No ex cubes minus X squared minus minus two X squared or the minus minus A plus, so that effectively that's minus X squared +2 X squared just gives us X squared. Minus six X minus minus 8X. Well, that's minus 6X Plus 8X gives us plus 2X and bring down the next one. X squared plus 2X plus a 12 X squared goes into X squared once. And so X squared plus 2X plus eight. And again we can see these two are the same when I take them away, I will have nothing left and so this is my answer. The result of doing that division is that. Well, the one that started us off on doing this was if you remember. X cubed minus one over X minus one. This looks a little bit different, doesn't it? Because whereas the space between the X Cube term and the constant term was filled with all the terms? This one isn't. How do we cope with the? Let's have a look. Remember, we know what the answer to this one is already. So what we must do is right in X cubed and then leave space for the X squared term, the X term and then the constant term. So what I asked myself is how many times does X go into XQ, and the answer goes in X squared. So I write the answer there where the X squared term would be. X squared times by X is X cubed. X squared times by minus one is minus X squared. And subtract X cubed takeaway X cubed no ex cubes. 0 minus minus X squared is plus X squared. Bring down the next term. There is no next term to bring down. There's no X to bring down. So it's as though I got zero X. There was no point in writing it. If it's not there, so let's carry on X in two X squared that goes X times. So record the X there above where the X is would be. Let's do the multiplication X times by X. Is X squared. X times Y minus one is minus X. Do the subtraction X squared takeaway X squared is nothing. Nothing takeaway minus X. It's minus minus X. That gives us Plus X. Bring down the next term. We have got a term here to bring down it's minus one. How many times does X going to X? It goes once. Long times by XX. One times by minus one is minus one. Take them away and we've got nothing left there and so this is my answer X squared plus X plus one, and that's exactly the answer that we had before. So where you've got terms missing? You can still do the same division. You can still do the same process, but you just leave the gaps where the terms would be and you'll need the gaps because you're going to have to write something. Up here in what's going to be the answer.