A LEVEL MATHS: Algebraic Division and Factor Theorem LESSON 2

hey guys just to let you know that you can catch these lessons live every Thursday from 6PM until 7 PM UK or BST time and I'll just allow you to ask any questions that you want and get some answers to them without further Ado let's get on with the lesson okay welcome to the second a level maths uh free tuition lesson I'm doing Live on YouTube So today we're going to cover the next part algebra that we need which is the factor theorem and algebraic division which you may be covering currently at school so let's start with what the factor theorem is so with the factor theorem it's pretty much solely used to figure out how we can divide a algebraic fraction so what they normally say is if X so you say something about if x um f of x plus minus a equals zero then X equals a or you can say X plus a Plus is a factor of f of x that's how they normally write it in textbooks right so if you want you can ignore this line so if f of x plus or minus a is zero then X plus or minus a sorry minus plus I don't think you have to flip the sign is a factor of f of x so that kind of doesn't really make too much sense right but let's say if we have something like um if we look at this first question we have a six marker question okay so it says use Factor theorem and division to factorize f of x completely so again this is an exam question so let's go through it so what you need to do first is you need to figure out what we need to do to divide this thing here because with a cubic I can't factorize it from scratch unless X is in every single term which it isn't in this case okay so what I need to do is I need to figure out how I can divide or why I can divide this thing by so what I need to do first I need to figure out using Factor theorem what will divide this function without a remainder okay and so what we do is we simply substitute in values so let's take F of 1 for example if I sub in F of one I get two times one cubed minus seven one squared minus 17 1 plus 10 which if you work it out again I would just use a calculator so this would be the quickest way to do it and what I would do is once you've typed it into a calculator leave it typed in and just change the number inside the bracket I'm going to write down a few right but just for a few examples but realistically you can do the whole thing on your calculator to find out what the factor is if I press equals I get Negative 12. okay so what I want to do next is I'll say okay what about increasing this to F of 2 what would that give me so now I'm going to go through using the arrows on my calculator and change the one that I put in all the ones I changed them into twos and I get negative 36 . that's getting further away from zero so now I'm going to start going okay what about F of zero well F of zero I can just see was going to give me 10. it's a bit closer what about F of one so let's change this to sorry negative one let's change this to negative one again all I'm doing is using a calculator and what you do in the exam is you'd literally just go through sometimes they give it to you which is lucky sometimes you just have to brute force it on a calculator I get 18. okay so this is getting closer to what I need so what about F of minus 2 . if I type in F of minus 2 and again I'm actually going to write this one out because I have a feeling this is actually the correct answer minus seven and this is all I'm typing into my calculator right make sure you use brackets and I get an answer of zero I'm willing to bet zero Okay so now what does this mean from the factor theorem okay so if f if F of minus 2 equals zero then X plus 2 is a factor of our expression f of x this thing up here and if we remind ourselves what a factor is a factor is a number that will divide into another number so if we said what are the factors of 16 well what can I divide 16 by well I can do 16 times 1 I can do 4 times 4 I can do 8 times 2. right so all of these are factors so with algebra obviously we don't just have numbers we also have a letter involved as well okay so does anyone have any questions about that thus far perfect so if X plus 2 is a factor of f of x what that means is I can now do algebraic division okay so what I'm going to do here is X plus 2 goes on the outside and we're using bus stop method okay exact same way we do long division with um numbers we do a very very similar thing okay now I just want to point out and if you notice all I'm doing is I'm actually typing in the algebraic expression exactly as it is minus 7x squared minus 17x now it's hard to just read through a mark scheme on this because the top row is going to change over time so you need to follow along very carefully what you do is you take the first term here and the first term here and you divide them so what's 2x cubed divided by X it gives me 2x squared then I take that 2x squared and I times it by both things in the divisor so 2x squared times x gives me 2x cubed again and that should always happen okay you should always get the same thing then at times about the other thing 2x squared times 2 is plus 4x squared okay and then what I do is I subtract both lines of these okay just subtract both lines of those so 2x cubed minus X cubed is zero minus seven x squared minus four x is minus 11x squared and it's also really important that you remember the minus comes in between so there are some issues sometimes when people subtract but then they don't uh carry the signs correctly and then the minus 17x and then the plus 10 is obviously exactly the same okay so now this is nothing anymore we do the same process again okay the exact same process that we just did okay so we take this 11x squared and we divide it by the first thing in the divisor so minus 11x squared divided by X is minus 11 x and then what do we do we times the minus 11x by everything in the divisor again so we're just repeating the same process again and again and again minus 11x times x is minus 11 x squared and remember I said that the first thing should always be the same okay because we've divided by it and then we're timesing by again now minus 11x times 2 is minus 22 x okay good then we do another line and we subtract them now this is where the subtraction comes into play you have minus 11x squared minus minus 11x squared so minus and a minus X Plus so this is minus 11x squared plus 11x squared which is zero and again these should always cancel out if it ever gets to the point where it doesn't cancel out then you've made a mistake minus 17 minus minus 22 well that's minus 17 plus 22 which is 5. and obviously the 10 is just carried down so these are both just carried down because there's nothing underneath them so we're subtracting nothing so far is anyone stuck or does that all make sense to everyone okay so now we've do the exact same thing again 5x divided by X is five yeah five times x is 5X 5 times 2 is 10. subtract again and you should get a remainder of zero if you get a remainder of zero you are now finished okay so what was the point of all of this well in regular long division the thing on top of the bus stop is our answer which in this case is exactly true so our value now is you write the divisor in a bracket X plus two and then 2x squared minus 11x plus 5. and you should notice that this will equal if we were to expand it again equals 2x cubed minus 7x squared minus 17x plus 10. it's the same thing on both sides of the expression so this is correct we're doing this well so far but now it says please what doesn't say please actually what am I talking about it says use this to factorize this completely okay so to factorize this completely so what we need to do is we need to factorize this quadratic we factorize quadratics a million times in GCSE so it should be okay we're looking for two numbers that times to AC which is 2 times 5 which is 10 and they must add to um B which in this case is minus 11. okay so you can just go through and you can work those out or you can kind of just using your deductive skills work out what you need to do there okay so is anyone unsure of that I think that should be okay either way once you factorize this you should get X plus two and you get 2X uh both one second two x minus one x minus five if you expand that that should give you the exact same answer as we started off with okay so again the best way to learn this topic is to just keep practicing so it shouldn't make that much sense yet you should make a bit of sense then as we do more and more questions it'll make more and more sense okay but these are all exam questions okay so that's a six mark question so it is meant to be a bit tricky it's meant to be a bit long but hopefully you see it's actually not that bad okay so let's go for another one then so this is a more typical style of exam question okay where they can essentially tell you what the factor is okay so for part A here we have to show using the factor theorem that X plus 3 is a factor of G of X well go back up to the factor theorem right if we substitute in a value of here and it equals zero then it's a factor so they're telling me that X plus 3 is a factor the way I show this is if I do G of minus 3 I need to flip the sign here and then to show that we actually need to do the substitution so 4 times -3 all cubed minus eight minus three all squared minus 35 times minus 3 plus 75 again this is kind of long so we need to do is substitute that into your calculator okay so again make sure you're using the chat to ask any questions that you may have because um again that's the only way you can learn so treat this like an actual lesson um there should be no kind of fear or anything like that because no one's going to know who you are anyway so don't worry about sounding silly yeah substitute this into a calculator just to double check that we've done this a-okay and it does indeed equal zero okay make sure that you check though because if you have substituted it in wrong we've done something incorrect you can change it and then get the two marks if let's say for some reason I substituted in plus three and I just said that equals zero I get zero marks okay whereas if I at least substitute it in I get one Mark so you get one mark for substituting in anything and then you'll also be able to see if you're right or wrong okay it then says hence sure that g of X can be written in the form G of x equal to X plus three X plus b squared where A and B are constants to be found so what they asking us here well all they're asking us to do is factorize it that's all they're asking us to do so all we need to do is do the exact same thing as the previous question we're going to do long algebraic division Okay so we put our Factor on the outside uh that's not actually put the bracket X plus 3 bus stop method and then all of G of X at 4X cubed minus ax squared minus 35x Plus 75. okay and once again uh what I'll do is I'll take this very slowly okay but please stop me the second that you are unsure okay so 4X cubed you divide it by it so you take the first thing here and you divide it by the first thing in the divisor 4X cubed divided by X gives me 4x squared if you notice I also line up so all of the X cubes go here all of the x squared all of the X's okay make your work nice and neat so it's easier for the examiner to read and so you don't make mistakes then I times it by everything in the divisor again so 4x squared times x and if you remember what I said if you've done this right this these first two things will always be the same 100 of the time 4x squared times three gives me plus 12 x squared okay and now what we're going to do is subtract these 4X cubed minus 4X cubed is zero then you have minus eight minus 12 which would be minus 20 x squared okay and then the rest of it is just dragged down because there's nothing else underneath it minus 35 X Plus 75. next step we take the first thing in our new line and we divide it by the first thing in the divisor so minus 20x squared divided by X which is minus 20 x okay now we times it by the divisor again minus 20x times x is minus 20x squared and once again these first two things should always be the same if we do minus 20x times 3 which is minus 60x once again all we do is subtract okay we just repeat the same process again and again and again until we can't anymore minus 20 minus minus 20 is minus 20 plus 20 which is zero minus 35x minus 60x is minus 95 x and again with the 75 it's just going to drag itself down because there's nothing okay let's give you a second just to make sure that you're 100 happy with that actually there is a mistake I made and just while I said that you guys shouldn't make this mistake I made the exact same mistake it's a minus 35 minus minus 60 would be minus 35 Plus 60. okay so that would be 25x okay then once again we divide it so 25x divided by X is plus 25 and then we times it again so we get 25x Plus 75. subtract it and we should get zero okay if we get zero that's a good sign okay that means we can actually continue with the question if you didn't get zero then go back and see if you made a mistake like I did and see if you did the minus incorrectly and things like that so now what do we do well this top bit is the answer to what the division would be so we can write X plus 3 and we've got 4x squared minus 20x Plus 25. and that will equal the exact same thing as our question which is exactly what we want okay so now we need to factorize this so again if we label this a b c you're looking for two numbers that times to AC right which in this case is 4 times 25 100 and adds to B which in this case is minus 20. okay so whatever you think those numbers are or again you can use some of your intuition to work out what those would be so again whichever way you prefer to factorize is perfectly acceptable but you should find that you get 2x minus 5 to x minus five but have a look at the question it should be something squared so now we just make it X plus 3 and we have a repeated bracket so we can make it 2x minus five all squared okay so we went through that pretty damn briskly any questions about that at all okay that's perfect so hopefully you're seeing how algebraic division can work and can be useful yeah perfect makes sense and yeah hopefully you can see how the factor theorem also works in this regard okay now we have the next one so this is another type of question that they can ask you obviously but they give you two pieces of information and if you notice you have extra unknowns in your original cubic right so we have PX and Q which means we can't factorize it until we know QX oh sorry p x and Q R but they've also given us this piece of information they said f of four equals zero so if we were to sub that in right now we would get 4 cubed 6 times 4 squared plus P times 4 plus Q should equal zero that's what that information tells me okay so 4 cubed uh how do you know that's 64. and then honestly you can just do the whole thing in your calculator anyway but I'm just going to work out what the numbers are so 4q plus 6 times 4 squared is equal to 160 plus 4p plus Q equals zero and they also give me the information F of minus 5. should equal 36. so once again let's go minus 5 cubed plus six times minus five squared plus P times minus five plus Q equals instead of zero equals thirty six so again let's work out what these numbers are and you can just use a calculator because it's a calculator paper but make sure you're putting everything in Brackets exactly as I've written it otherwise you might get the wrong answer that'd be 25 minus 5p plus Q equals 36. cool what might help is if we put all the numbers to one side for both of these Expressions so let's do that quickly and for this one that would be minus 5p plus Q equals 36 minus 25 is 11. okay so we have two unknowns and we have two equations so we can do simultaneous equations okay so if you ever hear two unknowns and two equations it's always simultaneous okay so we can label these one and two and I can do something with one and two to make one of the unknowns disappear so if I look at these two equations I have 4p and Q and I have minus 5 p and Q Q is common between both of them so I can subtract these two equations and that will give me one variable one unknown so if I do one minus two it doesn't matter which way around you do it we have four P minus minus 5p which is nine p yeah Q minus Q is nothing so that equals with minus 160 minus 11 is minus 171 so now all we need to do is divide both sides by 9. which is minus 19. okay is everyone with me so far okay so I think that's fine okay perfect so now once we get one variable in a simultaneous equation we can work out what the other one is by substituting it in to either one of the Expressions so if we look at equation one we can now substitute in our value for p so we have 4 times minus 19 plus Q equals negative 160. so 4 times minus 19 would be minus 76 plus Q equals negative 160. so Q is equal to minus 160 plus 76 which is negative 84. and that is our answer for part a there's only three marks so the marks in terms of where you get them is you get a mark for getting p and Q correct obviously and another Mark for setting up a simultaneous equation that's where you get your three marks from so I'll let you just have a second to think about that and see if it makes sense foreign so now we can go into Part B so Part B factorize it completely well again what you should know is it's a cubic so we need to use algebraic division so we're going to use the fact that F of four is zero so what we're going to divide by is x minus four remember we have to swap the sign in front of the number okay and then we're dividing what we're dividing we're dividing x cubed plus 6X squared P was minus 19 so minus 19x Q was minus 84. okay and now we can actually do the division so x cubed divided by X gives me x squared times about everything in the divisor again so we get X cubed minus 4x squared here and then we're subtracting X cubed minus X cubed should always equal zero so that's a good job I'm going to stop writing the zeros but let me know if it uh confuses you a bit and then we have x squared times -4 which would be sorry we're doing 6X squared minus minus 4 which would be 10 x squared right because the minus and minus becomes positive and these two things because there's nothing underneath them we just drag them down cool then again we start the process again divide so 10x squared divided by X gives me plus 10x 10x times x gives me 10x squared and again the first thing should always cancel 10 times minus 4 is minus 40x there we are subtract again 10x squared minus 10x squared is again 0 minus 19 minus minus 42 would be minus 19 plus 40 which would be 21 x and the 84 should come straight down again divide so Plus 21. times the again 21x times or minus 4B minus 84. all cancels out we're good to go so what we're factorizing is x minus four and x squared plus 10x Plus 21. okay and we're looking for two numbers that would times to give 21 and add to give 10 so hopefully you are fine with that is anyone unsure of how to do that so in this case two numbers at times 21 add to 10 would be 7 and 3 so we can just shove them straight into brackets okay are there any questions so far okay perfect so again very similar question okay except it goes a little bit further for Part B just a little bit so let's get to it First Step I'm hoping should be relatively easy for you to do relatively and all we need to do here is back to theorem so if x minus 2 is a factor then f of x equals 2 should equal zero so again you need to do the substitution first it's also a good way to check if you've done the question right right and see what that equals so again let's type in our calculator 2 times 2 cubed minus whoops 2 squared minus 13 times 2 plus 14 to suffer that in and you do get zero easy two marks so if Part B hence or otherwise so again if you remember from our previous lesson the word hence means using part a okay so this is obviously an important piece of information so what we're going to be using is the fact that x minus 2 is a factor to do as we've done a thousand times so far long algebraic division okay so again take the first thing here divide it by the first thing in the divisor 2x cubed divided by X is 2x squared then Times by everything in there so 2x cubed minus 4x squared and then subtraction you get zero minus x minus 1 minus minus 4 will give me plus three x squared and these things come straight down and we get minus 13x plus 14. divide it again we get plus 3x times it again 3x squared so you can see how quickly you'll be able to do this eventually slowly speed up to build confidence zero minus 13 minus minus 6 would give me minus seven X and then the 14 will come straight down divide it you get minus seven and times about minus two you get minus 7x and you get plus 14 which again if you subtract would give me zero so my equation has now become x minus 2 and we get 2x squared plus 3x minus seven okay and we want to solve that that equals to zero now they give us a hint two decimal places so I know I can't factorize this right because it's going to give me a decimal so what can I do instead well what I can do is if this bracket here equals zero then the whole thing equals zero right so all I need to do is solve this and we're going to be using the quadratic formula minus B plus minus the square root B squared minus 4AC over 2A where a is equal to 2 B is equal to 3 C equals minus seven okay and you should get two values of x okay two values of X you get are 1.27 after you sub it in rounded to two decimal places x equals minus two point seven seven now is that it in terms of our Solutions the answer is no this is a cubic which means it should have three solutions so we also have a solution from here which is that x equals two in fact we did that in the first part so the cubic there should be three answers the answers could all be the same but there should be three answers nonetheless so just keep that in mind if you have a cubic you should have three answers the answers can be the same so there might only be two unique ones but there's definitely three answers in total and again that's including that first bracket those are all the possible values of x such that this whole thing equals zero I went through that pretty quickly just to show you what can be done how fast you can get these things done but anyone that's unsure okay perfect so let's go on to the next one and again as you can see the question is getting a bit more simple in its language part a given that x minus 2 is a factor of f of x find the value of K so I'm subbing in F of 2 and that should equal zero so let's start out 2 cubed is eight plus two k minus two zero so eight minus two is six minus three so just move the six over to The Other Side by minusing six and divide by two and then hence it's saying to solve the equation f of x equals to zero cool so what we're going to have is x minus 2 on the outside because once again we're doing algebraic division and then we have just the equation so X cubed instead of K we're now going to put minus 3 because we know what it is minus two okay x cubed divided by X gives me x squared so in this case it can be tricky to right read the answer I'm just going to put it right above because we don't have any squares in this case times it by the cubes we get X cubed minus 2x squared and if we subtract it what might have helped in this situation is if you'd like if it skips over one of the things you can always put plus 0x squared minus 3x minus 2. okay subtract that you get zero minus minus two so zero plus two which is two x squared and these bits just get dragged down because there's nothing underneath them minus three x minus two divide by X you get minus two x times it again you get uh hang on a minute that should be plus two x squared and then minus 4X again subtract it so you get zero minus three minus minus four gives me X and that'll be minus two divide it you get one one times x is X and then you get minus two subtract you get zero just so you know you don't always get zero okay so if I had a remainder here let's say of three okay that is possible and one of those questions is coming up okay so if you want to see what that looks like just stay around for a little bit longer and it's saying for me to solve this now so x minus 2 x squared plus 2X plus 1 equals zero so we need to factorize this a little bit here or use the quadratic formula whichever one you prefer but we want so if this is a b c we're looking for two numbers at times two one and add to two so the only numbers there are X plus one and X plus one right so just one and one so in other words we have x minus two actually if I leave it there for a second hold on hold on I just want to prove something to you our Solutions now are going to be whatever the opposite of these brackets is so x equals two x equals minus one and x equals minus one there's always three solutions but because one is repeated we don't have to include it but there's still three answers it's just two of them are the same okay if that makes sense push X so again six marks should have taken the six minutes it only took us about three and again that's with me somewhat explaining what I'm doing so this isn't a bad topic but if you don't practice this you will make a mistake in the exam because very easy to make a mistake as I did earlier okay incredibly easy okay so once again it's the same question but different wording so we have this equation here use the factor theorem to show that x minus 3 x equals minus 3 is a solution okay so slightly different wording but all they're asking me to do is use the factor theorem to show that this is true again just flip the sign okay so let's sub it in okay and then just use a calculator to sub it all in and you should get zero and then again they change the wording instead of solve the equation or find the solutions they say find the other Solutions but that's only because we have one solution one solution is x equals minus three yeah but it's the same thing so all we need to do is do that X um plus three and do the algebraic division yet again by the end of this you should be bored of this so if you're getting bored that means you're learning that means you're understanding it's getting easy if you aren't bored it means you're still learning okay so just make sure you stick around for a bit longer X cubed divided by X gives me x squared times it again I'm gonna try and get through this slightly quicker 3x squared subtract 2x squared divided by X and then we can put the 4X here once again okay just to see how fast we can get it done minus five x and then there'll be a plus five on the end uh that should be at 15 actually it is is a 15 it looks very very not like one and that'll be minus five and you get minus five x minus 15. goes down to zero a-okay so that's how quickly you can do those divisions okay but again don't Focus that much on speed they give you five minutes worth of time to do that which is a long time anyways we have x uh plus three and we have x squared plus 3x minus five equals zero and again we're looking for two solutions now again remember the hint they gave us they've given us the hint that it gives it we want answers to two decimal places right if I look at this are there any numbers I can think of that times to minus five and add to 3. no because they're long decimals so once again we're going to be using the quadratic formula to answer this a equals 1 b equals three C equals minus five does anyone need me to use the quadratic formula and go through it once or are you all okay I think you might be okay but I'm not sure should be okay right so again if you just type the whole thing into your calculator so give me a second to do that you should get x equals 1.19 again it says the two decimal places minus 4.19 and we don't actually have to include the x equals minus three because the wording of the question says other Solutions okay so again read the question carefully but again they're just asking you to do the same thing over and over and over again okay so now we're coming on to a few more of the exam questions we're actually approaching the end uh this does come up every year but we've but only usually comes up maybe once or twice right so in the whole year which means there aren't that many questions on it now if you have your textbook obviously you have a fair few more but I'm only going to be doing the exam questions for now if you do have a textbook question that you want me to help you with the best thing to do is to in the in the description of this stream I have an email address you can email me any questions you have and I'll get back to you okay I'll give you a written solution for that or I'll make a video on it and then send you the link to the video whatever okay anyways prove that x minus 4 is a factor of f of x I don't know why I wrote four cubed minus 119 48 and once you set that in you get zero so you might have noticed that there's a part A but no Part B right there's a reason for that there is a Part B in the exam paper but it's actually Beyond this lesson so you're going to be using this again in the future when you do integration and differentiation okay so that's why it's not here at the moment because we haven't covered integration and differentiation and I'm not going to for a little bit so there's no point in me including Part B C and D for this question because you wouldn't be able to do it okay but just so you're aware this comes back in Future topics okay and this question actually isn't that bad right so if X plus 3 is a factor then that means F of minus three equals zero and that's due to the factor theorem okay so if anyone else with the factor theorem is you should know how to explain it so let's use that information so we have three times minus three cubed 3 cubed is 27 plus 2 a times minus 3 squared minus 4 times minus 3 plus 5a equals zero so let's work out the numbers okay so we have three times minus three cubed that gives me minus 81. minus 3 squared is nine times two is eighteen minus four times minus three is plus 12 plus 5a equals zero gather the A's together eighteen plus five is twenty three a minus 81 minus 81 plus 12 will give me an answer of minus 69. equals zero so move this over to The Other Side by adding it you have 23A equals 69 so a is equal to 69 over 23 which is 3. and that will be our answer so again there's a few ways they can ask you to use the factor theorem give me a second just to copy that down any questions at all perfect yeah good to see okay so we do have a few more before I go on does anyone have any questions about any of that sounds like it's all okay so what I'll do is I'll do the next two questions uh stop me if it doesn't make sense so use a factor theorem to show that X plus two is a factor so hopefully you're seeing a little pattern here that these are always the same okay [Music] it doesn't need equal zero and hence show that can be written in the form okay so we have X plus two and then we're dividing 4X cubed minus 12x squared minus 15x Plus 50. 4X cubed divided by X gives me 4x squared 4x squared times x gives me 4X cubed 4x squared times 2 gives me 8X squared and then let's do the subtraction minus 12 minus 8 will give me minus 20 x squared these bits will come down so we have minus 15x Plus 50. divided by X you get minus 20x that should be the exact same as it was before 20x squared because it always is times it by 2 you get minus 40x once again we're going to do subtraction minus 15 minus minus 40 would be plus 25 x and that 50 comes straight down so we have divided by X you get plus 25 times what x you get this do the subtraction you get zero so in other words you have X plus 2 x squared minus 20x plus 25. okay which is actually one that we've already factorized previously if you remember so if we go up and find that we don't have to redo the whole thing uh where was it yeah 2x minus 5 and 2x minus 5. oops so X plus 2 x minus five and since we've got it twice we can just put it as a little square okay so that pretty much covers algebraic fractions and the factor theorem so again it was a kind of annoying long tedious part but like I said it does come up in future topics for a level maths okay so things like integration differentiation partial fractions all of that fun stuff so uh we had to cover it and sometimes math is just all about doing 500 questions okay so I'm fine to stop it here I thank you all for joining in and having a look and just for those of you watching on YouTube later these are all live every Thursday 6 p.m to 7 P.M so the benefits of joining is that you do have a direct line of communication to me you can ask me any questions ask if you're stuck you can also send me any suggestions you have so certain questions you want me to do topics you want me to do things like that which is always fun and useful um but feel free to comment if you have any suggestions or you have any questions that you want to do other than that I'm going to wish you all a wonderful evening and I hope to see you in the next one