College Algebra - Lecture 4 P2 - Polynomial Expressions

now it's on to dividing polynomials we've multiplied we try to unmultiplied factor them and now we're going to look at what happens when you divide them and what happens is you will get expressions that are called rational expressions so we need to define that and then we'll also talk about the actual division of a polynomial by another polynomial but let's look at that first thing I want to define is what I mean by a rational expression a rational expression has a particular meaning a rational expression has the form remember we're always talking about forms and standard forms and mathematics has the form and here it is it's very easy polynomial over polynomial that's it it is not a rational expression if you have something other than a polynomial over a polynomial but the polynomials need not be the same there could be different degrees the top polynomials degree could be higher than the bottom the bottoms could be higher than the top or they could be equal all those possibilities exist but this is the basic form a polynomial over a polynomial so let me show you some examples just to get this in your mind the examples both of rational expressions so these are rational expressions and of ones that are not rational rational expressions look like oh for example x cubed minus x squared plus X plus square root of 2 over X minus x squared plus 1 I see there's no problem with square root of 2 because that's a constant that's a real number if I had square root of X that would be different because then I'd have X to the one half power and that would be something that's not a polynomial remember polynomials only have positive integer powers or zero so there's an example of a rational expression here's another one nice simple 1 1 over 1 minus X here's another one 5 X to the 4th minus 7 plus 7 you might say well wait a minute there is not a rational expression I don't see a polynomial over a polynomial well this is a polynomial and if you think of it as being over one which is another polynomial you'll see now that every polynomial is automatically a trivial rational expression now we've already attacked polynomials so we won't talk about them in this section we want to talk about the rationals expressions that actually look like rational expressions but remember polynomials are indeed rational what are some examples of not rational expressions not rational expressions well I just have to violate the definition of a polynomial somewhere for example I could look at as I said before square root of x minus 1 over X plus 1 now that certainly is a fraction with one function we haven't talked about functions yet but one expression in x over an expression in X but it's not a rational expression because the top is not a polynomial here's another one 1 over 1 minus 1 over X now you might say there's a rational in there sure 1 over X is rational but 1 over 1 minus 1 over X is not rational it may be reduced to a rational after simplification but as it stands it's not a rational expression because the bottom here 1 minus 1 over X is not a polynomial alright so there are a variety of things that looks kind of rational that seem to suggest that after some work they might appear rational like this last one but as they stand they're not rational so we will be wanting to talk about the degrees of the polynomials on the top and the bottom of rational expressions so let me give you a piece of notation that I will use quite often for example for the phrase long phrase the degree of say 7x to the fifth minus X cubed plus X minus 2 for that phrase a common notation is the following it's hard to say that remember also what the degree of a polynomial is it is simply the number that is the highest power of X attained in the polynomial the common notation for this is deg standing for degree of the polynomial you write the polynomial in parenthesis so the formula the format rather is deg of something all right in this case the degree of this polynomial is the way we write this that's just a common notation it'll simplify some things and just for the record here that number is 5 okay well let's go back and remind us of what we know about rational numbers so let's do a recall here rational numbers and then we'll see how that's going to extend to rational expressions so rational numbers come from the set that we've labeled Q now what about these they can be one of two types they can be proper rational numbers like what well three-fourths one-fifth seventeen twentieths etc and what is it that makes these proper the top is strictly less than the bottom the top number is smaller than the bottom number that's what makes these proper fractions or proper rational numbers or they can be improper as for example for thirds or seven halves or one hundred over eleven etc and what makes these improper oh I guess I should add another one five over five that's considered improper also what is improper about these the top is greater than or equal to the bottom so you have the two different kinds of things if the top is less than the bottom they are proper rational numbers they're improper if the top is bigger than or equal to the bottom now when they're improper we often divide okay when these are improper we often do a division for example four thirds you slip that in down here is three point three bar if we do a division okay so very often oops not three point three bar excuse me it's just a one in there one point three bar all right so sometimes a fraction can be divided and we do that most often with improper ones because we want to have these in a form that's more proper with the integers separated out from the fractional part all right let's go ahead and look at how this would apply to rational expressions so now rational expressions instead of rational numbers rational expressions we'd like to represent the same way so they can likewise be proper but now we don't want size of the number to indicate proper we'll talk about the degree of the polynomial on the top of the bottom so proper ones would be like x over x squared plus 1 1 over X minus 1 etc what makes these proper is that the degree of the top polynomial is strictly less than the degree of the bottom polynomial okay and then of course we know what improper would be improper would be X cubed plus X plus 2 over x squared plus 7 for example or 3x to the 4th plus 1 over 5x to the 4th minus 2 etc in these the degree of the top is either greater than as in the first one degree 3 degree 2 or equal to as in the second one same degree 4 degree of the bottom so that's how we will distinguish proper from improper rational expressions we look at the degree of the polynomials on the top and the by now just as with rational numbers when rational expressions are improper which is usually the way we want to do this when they're improper we often divide so I will show you the algorithm for dividing one polynomial into another so here's what I want to do divide and I'll write it as a rational expression 3x cubed plus 4x squared plus X plus 7 all over x squared plus 1 now you see the degree of the top is 3 the degree of the bottom is 2 so this is an improper rational expression and I'd like to do the division I'd like to divide the bottom into the top ok there's a piece of terminology here a couple of pieces I'd like to mention while I'm here this you can call the top polynomial top poly just to abbreviate polynomial if you like and this is the top polynomial is which is to be divided it is the thing we are going to divide there's another name for that it's called the dividend polynomial now dividend is one of those older words that's lingered on but it is simply it simply means the same thing as this is the polynomial which is going to be divided this polynomial on the bottom which of course is the bottom polynomial this is also known as the divisor polynomial so it is the thing which is doing the dividing it is the divisor okay well with this in mind let's go ahead and do this division so here is the technique and actually this is an algorithm an algorithm is a list of steps for doing something so here's how this works we put this in the center of my page the polynomial to be divided is three X cubed plus 4x squared plus X plus seven the divisor is put out front like so and this should remind you of long division with numbers this is exactly the same process now the algorithm says I want to find something to put up here that will multiply by x squared plus one so that I can get rid of this first term by subtraction so for example if I multiply if I take 3x up here and multiply it by x squared I will have 3x cubed and then 3x times one can't forget that plus 3x now when you're setting this up you want to line this up under the same power of X term and if there's a place where there isn't one put a blank space so that you can spread this out correctly now I have 3x times 1 so that's 3x I'll put it over here so 3x times x squared plus one gives me this now the algorithm says take that and subtract it away now by design the first one 0 that's the way this works this term 4x squared is not involved with this so it simply comes down and then I have a minus 3x and an X so that's minus 2x and now I continue the process I ask myself what do I need to multiply by x squared to get 4x squared well that's clear plus 4 so 4 times that'll be 4x squared and 4 times 1 is of course 4 and I don't want to put it on the x line I want to put it over here where the constants go so 4 times 1 is plus 4 which I'll put over there then I will subtract this entire thing this entire polynomial from the one above 4x squared minus 4x squared is 0 by design the minus 2x simply repeats down here because I haven't done anything to it and then minus 4 from the 7 that's above will give me a plus 3 and now I've reached the stage where I have to stop why well this is a polynomial degree one X to the 1 power I have a divisor degree to anything I multiply by this will have a degree to at least so there's nothing I can multiply by this to get minus 2x plus 3 so I must stop all right what are all the parts of this called and what do I do with all of this once I'm done well let's go ahead and box some of these in because they all have names the one underneath as you recall is the dividend or the polynomial being divided x squared plus one is called the divisor or this object on top is called the quotient and it's the quotient polynomial they're all polynomials and on the bottom here this of course is called the remainder what's left over the remainder polynomial and the relationship between them is the following and you can see it if you have this algorithm sitting in front of you if you take the quotient times the divisor and then take the divisor take that product and add it to the remainder then what you will end up with equals is the dividend so it's a nice circular arrangement the quotient times the divisor plus the remainder equals what's underneath there which is the dividend I'll bring this picture back but let me write down what that means for us so as I just said that would be 3x plus 4 the quotient times x-squared plus 1 the divisor plus the remainder which is minus 2 X plus 3 that's going to equal 3x cubed plus 4x squared plus X plus 7 which was the dividend now this will have bearing later in the course we will actually discuss this in more detail let me put it in words quotient times divisor plus remainder always equals the dividend or the thing being divided and there's one other thing to note here we already know have seen that the divide that the remainder rather is a degree less than the divisor because if it were higher or equal to we could still divide in further so the degree of the remainder polynomial let me put that in remainder polynomial is less than the degree of the divisor polynomial okay that has to be true otherwise you could have divided further so how would I also write this final result this is one way to rewrite it to actually see the pattern that's going on but we can also write it in a way that you might be more used to the original expression was remember 3x cubed plus 4x squared plus X plus 7 all over x squared plus 1 now we can say that that's equal to the quotient 3x plus 4 plus the remainder minus 2x plus 3 divided by the divisor and that's probably the way you remember writing things when you did it for long division for numbers you have a quotient polynomial and then a fraction here now if the top had been a perfect multiple of the bottom there'd be no remainder but it wasn't in this case so this parts left over ok let me do one more example and then you'll have to try this on your own let me bring back the one we just did just for a moment there's the technique remember we did the long division as you did with numbers we got the quotient times the divisor plus the remainder equals the dividend nice circular nice circular algorithm there ok let's try this one more time just to make sure you fix it in your mind we want to divide and here is the situation this time 4x cubed minus 3x squared plus X plus 1 all divided by X plus 2 Advisors a little simpler this time so here's the same pattern I'm going to write 4x cubed the thing to be divided minus 3x squared plus X plus 1 underneath here my divisor is X plus 2 and then I proceed with the algorithm I will take 4x squared because 4x squared times X will give me my 4x cubed then I from this I'll get 8x squared plus 8x squared again the algorithm says subtract that away by design the first thing is 0 and then minus 8x squared and minus 3x squared is minus 11x squared bring my ex down plus X I'm ready to go again how do I get a minus 11x squared well I'm going to need a minus 11x here minus 11x squared minus 11x times X minus 11x squared times 2 will be minus 22 X I subtract that away that's what the algorithm says to do I have now 0 here by design and then minus a minus is plus 22 plus X is plus 23 X I guess I don't need the Plus do I and then I bring the last constant down plus 1 and I can still continue because the degree here is 1 that's the same as the degree of the divisor is not less yet so I can do it one more time plus 23 23 times X plus 2 is 23 X plus 46 subtract that away that is the algorithm here again I get 0 by design 1 minus 46 is a minus 45 and finally I must stop because that is of degree zero and that is one less than degree of the divisor which is degree 1 so again here is my divisor my quotient up here if I multiply it by the divisor and add it to the remainder then I will end up with the dividend I won't write that out this time I'll simply write it out the original expression remember was 4x cubed minus 3x squared plus X plus 1 so I will write this out in the form that you might be more used to this will then be equal to the quotient which was 4x squared minus x+ 23 then minus 45 over X plus 2 and that might be the form you've seen this in before now all of these forms have their uses this turns a rational expression into a nice polynomial and another piece here and calculus for example this is a very nice form because it has bearing on taking of what are called limits but for us right now it's simply a matter of division so here's a tip okay here's a tip whether a rational expression is proper or improper either way proper or improper either way having the top and bottom factored factored as nicely as you can having it factored is very useful okay that's something you always want to try and do why let me show you why with a couple of examples illustrate once again to continue my remarks the effort of either division or simplification as we'll see later whoops simplification the effort of division or simplification is easier and here's an example if you start out with x squared plus 3x plus 2 over X plus 1 if you know that the top factors into X plus 2 times X plus 1 over X plus 1 this is much nicer because X plus 1 over X plus 1 is 1 that of course will assume X is not equal to minus 1 so we're not dividing by 0 here but that's 1 right that's what one is so what are we left with X plus two now that is much nicer than the original expression and I couldn't have gotten there as easily if I didn't know that the top factor okay here's another example or X plus 1 over X plus 1 to the fifth now if you wanted to simplify that it's very easy to do X plus one over one of these X plus ones and again we'll assume that X is not minus one is going to give you 1 over X plus 1 to the fourth that's easy now let me pose the same problem to you X plus 1 over and instead of having it nicely factored let me give you the multiplied out form X to the fifth plus 5x to the fourth plus 10 X cubed plus 10 x squared plus 5x plus 1 now let me ask you to simplify that that's much more painful this of course is harder why because the structure of this is not at all clear up here the structure is crystal clear and the simplification is easy here it's all blown away this is someone who has succumbed to the multiplication disease and unless you're lucky enough to recognize this form as X plus 1 to the fifth which is doubtful at this time you have a much harder expression so if you have something factored like this do not multiply it out and if it's not factored try and factor it on that note it's time for us to pause and for you to go ahead and try a few of these well we've divided polynomials now looked at rational expressions and now we're going to pull a lot of this together into something I call the art of simplification because simplifying things is again one of the things you need to be able to do in mathematics because we want to make our calculations as easy as possible and simplification is not just a series of techniques you apply in a rote manner it really is a kind of art and so I'm going to show you a few things you'll have a chance to try some things and then we'll come back and see if my method in is the same as yours it doesn't matter which method you use as long as you get something simple so don't feel obliged to follow every last line that I do here okay the art of simplification let me start right off with some tips and after that we'll do some examples in which I Ellis trait some of these tips first tip I've already said before let me say it again factor whenever possible this usually means factor when you recognize that there's something you know how to factor okay here is another key idea which we've seen again and again the key idea is one from rational numbers if you have a times K over B times K guess what that is it's a over B why because K over K is one that's why classic idea but there's something lurking in here that you have to be careful about be careful k is a K I'll put it in quotes is a factor of the top and bottom not added not subtracted it is multiplied by a and multiplied by B so you get this one form that's the only way that this can work so here's an example of it working and I'll show you another one where it's wrong example two times X minus five over x times X minus five now if we assume that X is not zero so we don't have division by zero here and X is not five so this doesn't give us division by zero then this is equal to one which means this is two over X so that's a good use and a correct use of that one property here is an incorrect use incorrect attempt you might see a student do this to X cubed course you would never do that right to X cubed plus y squared over Y squared this is what they would say is equal to one and they would say equals 2x cubed plus one no that's not true this is not correct why is it not correct because Y squared is not a factor of the top that's the key see up here X minus five is a factor of the top the entire top and X minus five is a factor of the entire bottom here Y squared is okay on the bottom but Y squared is only a term on the top it is added to two X cubed it's not multiplied and because it's not multiplied this is just not true okay so you want to be very careful of that and you'll have a chance to try that out because I will give you some some examples in a little bit well let's continue on with my tips more tips okay here's one it's very very simple it turns out to make a lot of things go more quickly 1 / 1 / box equals box I'm assuming that whatever is in the box doesn't make division by 0 happen but this is always true you call this if you want to be colloquial you can call it a double flip or if you want to be more proper you can call it a reciprocal of a reciprocal if you flip something over and then flip it over again you're back where you started now if you see this form you can immediately rewrite it as simply whatever's in the box that can make your life a lot easier now here's another tip that I want to qualify ok because it is not always the best thing to do so don't feel obliged to do it here is one thing that you can do find the LCD now we haven't really talked much about this let me tell you what it is and then show you a few examples this is the least common denominator remember we're calling things on the bottom the bottom the denominator is the old term that has continued to be used the least common denominator for two or more added rational expressions and that of course includes subtracted ones so we won't worry about saying that separately find the LCD or finding sorry finding the LCD for two added rational expressions may help now sometimes it is more trouble than it's worth you end up spending so much time finding the LCD that the problem could have been solved the hard way more quickly okay what this requires it requires at the very least factoring that's why we studied some factoring earlier factoring the bottom polynomial that is to say the denominator okay now let me go back into the past where you've seen least common denominator for rational numbers let me do a very quick review recall LCD for rational numbers okay I will show you by example how that works and then what you can do in an example for rational expressions suppose you want to add something simple like 1/12 plus 1/10 okay now here is the motivation we'd like a denominator less than 12 times 10 which is 120 see we can always multiply these two together and get a denominator that will work for the two of these but suppose we'd like to get one that's a little smaller well the technique as you may recall is factor 12 and factor 10 down to their prime factors 2 times 2 times 3 does it 4 12 and 2 times 5 does it 4 10 and then you take only as many as you need to create the two denominators 2 occurs twice here so we're going to need at least 2 to s 3 & 5 each occur only once in all of these numbers so we take one of each now this will be the common denominator the first fraction requires that I multiply top and bottom by 5 to get the right denominator now so that will leave me with a 5 on top this one requires more I have a 2 and a 5 but I don't have the other 2 and 3 so I need to multiply top and bottom here by 2 times 3 so I'll have 2 times 3 here well 5 plus 6 is 11 and what is 2 times 2 times 3 times 5 that's 4 times 15 also known as 60 now you see the denominator is less than 120 that in fact is the least common denominator you could have so you've done the best you could for a denominator okay in the case of rational expressions things are a little bit different let me make one note so I don't have to keep right rewriting this later as I do my many examples will assume variables don't cause division by zero so we're going to lots of expressions with lots of variables in them we just will say that the variables will never stand for numbers that will cause the denominator anywhere to be zero so that way I don't have to say it repeatedly for each of the problems also most coefficients below most of the coefficients below are integers this is usually not true let's be realistic about this I don't want to mislead you most of the things if you see polynomials and rational expressions in practice chances are good that the numbers that are multiplied by the powers of X are not going to be integers however for the case of learning this and for ease of computation for us by hand in this course we will have a lot of things that have integers as coefficients but realistically that's just not true that's why it's more important to understand the ideas involved here not just what to do in case of integers so it's time for examples so here we go example another one of those artificially nice examples but it will serve to illustrate the use of the LCD the least common denominator for rational expressions suppose I want to add x over x squared plus 3x plus 2 plus 2x minus 3 over x squared minus 1 now once again I will assume that whatever XE is it does not cause this denominator or this one to be 0 for example I know that X will be never 1 or minus 1 because that would make this 0 okay well if I am lucky I should be able to see some sort of factorization for my denominators I'll rewrite my numerators or the tops and I will look at the bottoms now this is of that form we looked at earlier that ax squared plus BX plus C form so I suspect there's no constant out front I expect is going to be a nice pair of binomials here I will try and see if I can get one as I said this is nicely constructed so I know that I will have X's there to give me mine x squared and if I put 1 & 2 I'll get 1 times 2 - 2 and if you just put pluses in there lo and behold you get 1x and 2x which adds to 3x so there's a nice factorization frankly that's not going to happen very often what about this one well this one we recognize as remember this is the difference of squares I can get it slipped in here difference of squares right x squared - 1 you say but 1 is 1 squared so it's a difference of squares so this would be X minus 1 times X plus 1 okay now we have these denominators factored as simply as I can make them over the real numbers so now I can construct a single fraction and I will pick out what I need to create a least common denominator I just need as many times as these occur in each one of these factors X plus 1 occurs once here and once there so I only need one X plus 1 X plus 2 occurs here but not here so I need an X plus 2 X minus 1 occurs here doesn't occur here so there's only one of them but I do need that one then I look at this fraction here and I say to myself what's missing down here well it's the X minus 1 so I multiply top and bottom by X minus 1 on the top that gives me x times X minus 1 the bottom of course is the correct denominator then this plus comes here then I'm going to want to multiply 2 X by minus 3 times whatever's required here well what's missing from these 2 X plus 2 is missing so X plus 2 top and bottom means that the top gets an X plus 2 and then let me continue this on to another page continuing with my equal sign on the top well the bottom of course is the same X plus one X plus two X minus one the top there is nothing common if you look back what I had a moment ago there's nothing common here that I can factor out so I just have to multiply everything out this is a case in which you really have to do that so I will do that I will get then x squared minus X plus two x squared plus four X minus three X minus six so in the second term here I distributed the two x and then the minus three okay now I can combine like terms on the top we've talked about that before let's see we have an x squared and a two x squared so I have 3x squared and then I go down to the next power I'm being systematic after all I look at the squares I look at the X is now minus X plus four X that's three X minus three X so that's good all the X's disappear what's left constants there's only one of those minus six so the bottom remains the same the top immediately can be simplified by factoring out a three the bottom remains the same and I would say that's a stopping point right here why am i stopping because although I could factor the top as you saw once before into x squared minus square root of 2 and X sorry X minus square root of 2 and X plus square root of 2 nothing down here is going to cancel with that so why don't I just leave this as it stands okay so I use the LCD the least common denominator to get this X plus 1 X plus 2x minus 1 denominator and in this case it worked out nicely but this is really an artificial problem I don't want to mislead you now in this example which again is a simplify question we have something that's much more complicated looking to minus 1 over 2 minus 1 over 1 minus 1 over X now there's a lot of internal structure here this is obviously not a rational expression but it does have rational aspects what we'll do is we'll start with the innermost point here and work our way outward turning everything into rational expressions as we go so first I need to repeat a lot of what is here before I can get down to where I want to operate so I have 1 over now down here I want that 1 minus 1 over X to become a single fraction with the denominator X of course so X will be the denominator I will multiply 1 by X over X and so it'll have the correct denominator and then X on the top will give me X minus 1 here but now I have something that I like to see I have 1 over a fraction which just says to me flip this fraction on the bottom over that's what 1 over does that was one of my tips earlier so continuing to minus 1 over 2 minus X over X minus 1 see this is the flipped version of the fraction let me bring back the previous page I had 1 over X minus 1 over X and now this flips over to X over X minus 1 which is exactly what I have here now I continue on this portion wanting to change it into a single fraction so I'll have 2 minus 1 over and here I want a fraction that has X minus 1 is its denominator X minus 1 well this fraction already does this part doesn't so I multiply to top and bottom by X minus 1 the top then becomes 2 times X minus 1 minus the X which was already there again I have 1 over a fraction that again makes me happy so I will that over so I'll have to minus X minus one large fraction over and let me pull back what I had before I had two X minus one minus X over X minus one this now is going to be flipped over that's why the x1 minus one is on top the bottom let's see what this is we see if we can simplify that to X minus X leaves me with an X so I'll put an X here and then two times minus 1 is minus 2 so that's minus 2 and now this already is much simpler than the original now I want to take the entire expression and turn it into one fraction all over X minus 2 well this already is over X minus 2 2 is not so multiply top and bottom by X minus 2 I will then have 2 times X minus 2 on the top minus the entire quantity X minus 1 be careful the - distributes over the entire top of that the entire numerator there and so I simplify the top here and let's see what I have 2x minus X that leaves me with a single X 2 - 2 x minus 2 is minus 4 plus 1 so I have X minus 3 over X minus 2 that's great that's much simpler than the original one so I'm very pleased at this point okay now here's what I want you to do we're going to pause and there are two problems that you'll be assigned now these problems I want you to work and see if you can simplify them what I will do when we come back is show you how I would simplify them so try the problems and I'll be back in a moment now you've had a chance to try two problems here's one of them here's how I would have attacked it X cubed minus 8 over X cubed minus 2 x squared the first thing I see is that the top and the bottom can be factored by familiar formulas the top is the difference of two cubes remember I see eight and I read two cubed so I have X cubed minus two cube we had a form for that so we will have X minus 2 times x squared plus 2x plus 4 so that has a recall you have to recall that that's the way that X cubed minus 2 cube factors the bottom is a little easier because I see there's a common factor of x squared I can pull out so I take that out what's left I have an X from the first term and a minus 2 from the second now I'm already feeling good because those are my X minus 2 on the top and the bottom and once again in this problem and on all the problems we will assume that the bottom is never 0 all right I have one other thing here the x squared plus 2x plus 4 and I ask myself could that factor well let me first of all establish that the X minus 2 over X minus 2 is 1 so I can remove that and I have x squared plus 2x plus 4 left over here and on the bottom I have just plain x squared now it turns out if you try to factor the top it doesn't factor simply and in fact it won't factor into any factors that we can do with our current number systems so that is where you'd have to stop not very satisfying perhaps but it's certainly a good place to stop now since this is improper if you wanted to you could go ahead and divide x squared into this and get a polynomial plus a remainder term but I'm not going to go ahead and do that the other example was this one this was a bit more complex just like the last one I did before we stopped 3 X minus 3 over x squared now that was the top all by itself the bottom was 1 over X minus 1 squared minus 1 now that looks terribly complicated but it isn't if you take it in a stepwise fashion for example I'm going to make the top and the bottom into single fractions then I'll have a fraction over a fraction and I know what to do with that so let me take that first step the top fraction should have x squared as its denominator so I'll write x squared here I already have an x squared denominator for the right term the left term doesn't so I multiply top and bottom by x squared that leaves me with 3x cubed on the top minus the 3 from here so now the top is a single fraction I'm going to do the same thing for the bottom here X minus 1 squared is the logical denominator the only one present so I'll take X minus 1 squared notice I'm not factoring it I'm not multiplying it out that would be a case of the multiplication disease I prefer to leave it this way then I need to multiply one top and bottom by X minus one squared so it has the correct denominator so on the top I will have one minus X minus one squared now I have a fraction over a fraction you remember how we do those I'll bring that back in a moment so you'll see first of all we simply copy down the upper fraction 3x cubed minus three over x squared and then remember what we did with the one on the bottom we multiplied and turned it over we flipped it over so this was X minus one squared all over one minus X minus one squared and still I'm resisting the urge you may have to multiply this out I like it better this way and you'll see why in just a second ok well now this isn't entirely into this is entire this is a single fraction I can just multiply the top and the bottom as I always do let me go ahead and do that and remember you can do more than one thing at once if you see it notice that here I have 3 X cubed minus 3 I see that there's a 3 I can factor I'm going to do that as I recopy this so 3 times X cubed minus 1 rewrites this upper expression times and here's X minus 1 squared which I'll just recopy the bottom the x squared is simple enough I'll leave it the way it is and this expression now look at this this is one minus X minus one squared here's what pays to leave it this way one can be thought of as one squared so this is a difference of two squares since it is I can factor it I'll use square brackets to keep things straight so that will factor into two factors one minus X minus 1 and 1 plus X minus one because that's how you do the difference of two squares now we can continue on and simplify even further 3x cubed minus one you may see as the difference of two cubes because one can be interpreted as one cubed so I could take X cubed minus 1 cubed and break that down into factors X minus 1 and x squared plus X plus 1 times the remaining X minus 1 squared which I've just copied over on the bottom I have x squared now what do I get here first 1 minus X and then plus 1 so I have 2 minus X and then the last term becomes even nicer 1 plus X minus 1 the 1 and minus 1 add to 0 so I just have another X well I'm getting to the point where I have lots of nice things happening let me recopy this on the top notice I have an X minus 1 here and an X minus 1 squared so I can combine these into X minus 1 cubed which I will do next so on the top I will have 3 times X minus 1 cubed times that other second-degree which I cannot change and on the bottom I had x squared and X which I can write as X cubed times 2 minus X and that is where I am going to stop because x squared plus X plus 1 does not factor in any nice way for me in fact it will only factor in complex numbers so I'm done there and there's nothing else that's in common here so I'm going to stop and I feel accomplished that is a simple thing to do now I'm going to pose you yet another problem this is a little different this doesn't have any rational expressions in it so you might expect that it's easy here it does have to do with a polynomial expression and polynomial ideas and here's the question where is the error so what I will do is write this out show you everything that I do and then leave it up here for a moment to for you to look at and then we'll stop and you'll try it on your own and then we'll come back and I'll tell you where the mistake is given that y equals x so there's the starting point now I'm going to multiply both sides by X so Y X then equals x squared multiplying both sides by X then I will add x squared to both sides so x squared plus y x equals well x squared plus x squared is 2x squared no mystery about that then I will subtract from both sides x squared plus y X I will subtract -2 or just subtract 2y X so that's equal to 2x squared minus 2y X then I will combine these two so I'll have x squared minus y x equals 2x squared minus 2y X now you just want to see that these are all legitimate operations I will now factor both sides factor an X out of this side I'll have x times X minus y on this side I can factor I see common 2 X so 2x times X minus y then I will divide out the X minus y so I have X equal to X and then divide out the X so that I have one equals two what's wrong there's obviously something wrong here because as far as I know one is not equal to two something is wrong in what I did here now let's just leave this up on the screen for a beat and you will have it in your notes also so you can look at this at this point but now we'll pause and turn the tape off you could try and see where the mistake is and when I come back I'll show you where it is okay I'm back now do you know where the error is let me show you where the error is every operation I did was correct until I got to this stage now what did I do at that stage I divided by X minus y that was my problem because if x equals y x minus y is 0 which means I divided by 0 and that is not allowed and if you do something that's not allowed even if it's snuck up on you like this then you can end up easily with a contradiction like 1 equals 2 so that's what's wrong here division by zero so when you're manipulating polynomials be careful that you don't get caught dividing by something like this when you're not sure what the letters stand for all right on that note we'll stop this section and I'll come back in a moment with the next section well now that we've actually looked at the art of simplification it's time to go to solving some polynomial and rational equations so let's get down to that first we want to start with the definition and the definition is an equation it's always nice to have a definition remember we had one way back when we talked about the language of mathematics it was a sentence in which the verb is equals but for us we want to be a little more specific an equation in one variable say X that's a convenient variable that we'll use has standard form and I'm going to make this very very straightforward standard form that looks like this expression in X equals 0 that's it that's what an equation looks like you can always push everything over to the left and you'll have some expression in x equals 0 or if you like zero equals expression in X now either way you look at it that's what you can reduce an equation to and from that we can talk about what it means to solve an equation so that's a standard form of an equation now let's talk about solving such a thing these solutions which of course I'm going to be abbreviating as I have before the solution or sometimes called roots of an equation or what well they're numbers they are the numbers that you can put in for the variable and get a true statement so a solution or root of an equation a roots of an equation are the numbers all of the numbers the numbers which when substituted in for X this is longer to write than it is to say substituted in for our variable X that's the one I'm using here yield a true statement so when you put in some number that is a solution you get a statement that's true you get something equals something else and that's true and of course the solution set is well what else the set of all solutions there I'm abbreviating solutions so their solution and solution set and what the equations look like that we're going to be examining oh they might look like X plus 5 equals 0 now I will have already pushed everything to the left but if it's not there you can always put it that way there might be a rational expression x squared minus 4 over X plus 1 equals 0 and so on ok we'll look at all sorts of forms like that now you have to realize that equations come in two basic categories two categories and these are these are idea categories two categories of equations there I am abbreviating equations again one there are equations which are true only for finitely many numbers finitely many numbers usually maybe one number or two number for the kind of equations that we end up looking at some small amount of numbers not infinite the other kind of equations which are the ones that are true for all numbers well all numbers for which both sides are sensible for which both sides of the equation are defined and these equations have another name they are called identities that's what an identity is it's something that's true regardless of what you put in for the variables provided you don't do something that makes one of the sides undefined so equations that are true only for a finite number of numbers are what you often see in practice identities of the kind of things that you see structure in that you see how the algebra works these are the things we saw before in our product formulas or our factoring formulas so these are the two basic types of equations now how do you solve an equation we're just getting all the words out here to solve an equation means you start with the original equation and you replace it by equivalent equations that are simpler so replace your equation by equivalent equations which are simpler that's why we spend so much time learning how to make things simple all right and what do you do to actually go from your equation to an equivalent equation what what are you using you're using algebra anything you know about algebra is how you get there and how simple do you want things to be you want them to be as simple as possible you would like to have something like x equals 5 now there's an equation that I can handle simple I know what X is that's the end of the story would that all of them were that easy all right well there is and I'll put it on its own sheet because it's so important there is a handy fact remember fact is a theorem here's a handy fact you might even give it a fancy name you could call it the zero-product principle that makes it sound exotic the zero-product principle here's what it says you already know this if a times B equals 0 then this is a theorem then a equals 0 or B equals 0 that's all that this says it says if you have box times box equals 0 then one of the two boxes has to be itself 0 now this makes equation solving much easier but it does require that the left-hand side be written as a product in other words that the left-hand side be in some way but when it is then each part has to be zero or could be zero to make this zero so this is really a very handy principle that you want to keep in mind alright let me show you how this works now let's go ahead and actually solve an equation here example solve for x in the following X plus one times two x equals x plus one times two here's my solution now let me warn you against one thing that students often do which is not good to do because it eliminates a solution that may be the only solution here's what people often try and do they look at this and say look X plus 1 is here there's an X plus 1 here why don't we divide out by X plus 1 well be careful don't divide by X plus 1 Y because this assumes you saw the problem we had earlier with the where's the error problem this assumes that X plus 1 is not 0 because that's the only way you can divide by X plus 1 well if X plus 1 is not 0 that means of course that X is not equal to minus 1 unfortunately x equals minus 1 might be the only solution to this and by dividing you toss this out so the better way is not to divide just simply pull everything to one side and to make this simpler I'll move this to another page I'm going to pull everything over to the right so I will have 0 on the left now watch what advantage this will give to me here's the original X plus 1 2 x equals x plus 1 times 2 I'm going to move this to the other side by subtracting it so I'll have 0 equals x plus 1 times 2 that was what was already there minus X plus 1 times 2x now it really makes no difference which side I put this on but by putting it on the right hand side I can continue my equal signs in what is effectively one long sentence there's no period here I can now continue by saying alright let's factor this X plus one is common in fact two is common also so I can factor out 2 times X plus 1 what's left from here one minus from here X so I have this now thoroughly factored and now I can put my period see this becomes a long sentence we'll see more examples of this a long sentence so that the 0 equals this and you have to imagine that this is coming round like this okay so 0 is equal to this the zero product principle by the zero product principle which I'm abbreviating X plus 1 equals 0 that's this term or 1 minus x equals 0 well X plus 1 equals 0 gives me x equals minus 1 see I told you this might turn out to be a solution or 1 minus X is 0 that means 1 is equal to X and so I now have two solutions what I should do is go back and check them in the original expression my original expression was X plus 1 2 X in fact let me write it down so I have it here X plus 1 times 2x equals x plus 1 times 2 and I ought to check that both of these solutions actually work minus 1 I will put in here minus 1 plus 1 times 2 times minus 1 does that equal minus 1 plus 1 times 2 well sure it does that's the first one that gives me 0 equals 0 that's ok and to check the other one I'll have to go to another page here the original equation remember is this X plus 1/2 times 2x X plus 1 times 2 go to put a 1 in there now so I'm checking 1 1 plus 1 times 2 times 1 is just 2 and then 1 plus 1 times 2 well of course they're equal so that also works now you might say well if I did the algebra correctly why do I need to check my solutions you will find out as we continue on that is very possible to do all of the algebra correctly and end up with something that cannot be a solution or to end up with more than one number sum of sum and one of them may not be a solution so we'll look at examples of that and we'll talk about them as they occur let us go ahead and look at another example so this is solved for X again here's the expression a bit more complicated 3x X over X minus 1 plus 2 equals 3 over X minus 1 and we'll make the assumption that X is not 1 now this is built into the problem why because I can't divide by 0 and if X were one I would have zeros here so I will assume that X cannot be 1 well since since X minus 1 is then not 0 I can multiply by it as follows multiply both sides of the equation by it so I will have X minus 1 times the left side of the equation remember was 3x over X minus 1 plus 2 and that will be the same that will be equal to X minus 1 times 3 over X minus 1 that was the right side of the equation so this was the left side and this was the right now multiplying through by X minus 1 I'm going to have my X minus 1 over X minus 1 here which is 1 so I have 3x plus 2 times X minus 1 because there's nothing to make that equal to 1 there equals again X minus 1 over X minus 1 is 1 so I just have 3 and now I have an expression that has no fractions in it no rational expressions much easier now I'll move everything over to the left here I have 3x and I'll simplify as I go 2 times X minus 1 plus 2 ex- to bring the three over minus three equals zero well now I'm on a roll 3x plus 2x is 5x minus 2 minus 3 is minus 5 equals zero factor out the five times X minus one equals zero by the zero-product principle X minus one must be equal to zero and there's only one solution for that x equals one and I feel just great until I remember what I said at the beginning remember the original problem this solve this where X is not equal to one why because we don't want to divide by zero X can't be equal to one the only solution I end up with is X equal one what is the conclusion conclusion this equation has no solutions because X minus X equal 1 is eliminated by definition at the beginning and X equal 1 turn out to be the only thing that would have worked so there are no solutions and that's a perfectly legitimate result not every equation has a solution all right so let me go ahead and enshrine that in a remark some equations have no solutions if you've checked your work and you check the results don't feel bad if you come up with that conclusion my second remark I'll put on the next page here my second remark is when you solve or solving by algebra which is using all the techniques of algebra to simplify things only yields candidates for solutions as we saw a moment ago one was a candidate for a solution which these candidates must be checked and they must be checked where in the original expression because that is after all what you're trying to solve that's the whole point the original unsimplified equation now that is the best advice I can give to you when you solve things by algebra sometimes there are no solutions the numbers that you get you can consider candidates for solutions candidates which must be checked if you want to see if the candidate is really a solution put it into the original unsimplified equation don't stop at any intermediate step put the candidate here into the original unsimplified equation and if it's true you've found a solution on that note we're going to pause you can try a few of these well now you've had a chance to work this problem let me show you how I would attack it all right the question is solve this equation X cubed equal 25 X and I put a period on the end because remember an equation is a mathematical sentence and I want to make a point of that as I'm working through here so solution alright the first thing I'm going to do is move everything over to one side your instinct as I warned you may be to divide both sides by X but this is not something you want to do because division by X assumes that X is not 0 x equals 0 may in fact be a solution and if you divide by X you'd lose that solution so the best advice as we saw before is to move everything to one side I'll move it to the cubic side so I'll have X cubed minus 25 x equals 0 ok again period that's the end of a mathematical sentence now as I'm looking here on the left I'm thinking ahead and saying to myself I have something equal to 0 if I had a product equal to 0 I could use that zero product principle I could have a product and I would know that one of all of the factors could possibly be 0 so I'm going to see if I can factor this well I see a factor immediately I see that X is common to both of these so I'm going to go ahead and factor that out what I'm left with is x times x squared minus 25 equals 0 again the end of a mathematical sentence and now I look at this and I say AHA I recognize this because 25 is a familiar number to me that's 5 squared and I have the difference of two squares so I'll write that down here difference of squares well with the difference of squares I can factor this nicely that's one of the forms that we looked at earlier so I have x times X minus 5 times X plus 5 equals 0 again the end of a mathematical sentence and now I'm prepared to finish this off because as I quoted earlier the zero product principle which applies to three factors this time okay this principle says that in order for the product of these three to be zero one or or all of them needs to be zero so let's go ahead and see the possibilities here we have x equals zero that's the first factor here or X minus five is zero that's the second factor or X plus five is zero well x equals 0 is a solution as it stands so we'll just leave it like that X minus five equals zero we can solve for X and get x equals five there's another candidate for a solution and X plus five equals zero leaves me with x equals minus five so what we have found here are three solutions x equals zero x equals five and x equals minus five now one of the things i told you about before was that you really ought to check your solutions i'm going to leave the checking to you here i won't have the space to do that so I'll just let you check but it's pretty easy to check if you think about it the zero obviously makes both sides here zero and the five and the minus five are also going to give you a true statement so I leave that to you but this is how I would have solved this and these are the three solutions to this now remember I noted the periods at the ends of these mathematical sentences the idea behind that is going to come up again okay in fact it's going to come up right now I wanted to make some notes about notation and let me just call this a a notation pattern now you'll see me doing this a lot you'll see a lot of people doing this and you need to understand what it means when people do this and not to read it incorrectly so I'll give you two examples first example suppose I'm with 2x to the fourth minus 32 and what I want to do is simplify it I want to start working on it and factor it and get it into a nicer form well I'm going to put an equal sign here notice I can factor out a 2 so let me do that first 2 times X to the fourth minus 16 and now I continue notice I put the equal sign below this is going to be part of the point I'm making also notice that I didn't put a period at the end because my sentence is not ending alright I continue to factor if I see more I see that this is a difference of two squares sixteen is four squared so I can rewrite this as x squared minus four times x squared plus four and then I see that this first factor is a difference of two squares and so I can go ahead and factor it a little further X minus two x plus two times x squared plus four now I'll put a period for the sake of this demonstration what have I done I've created one long mathematical because that's what we're looking at here sentence okay that's the point I want to make here this is one you can call it a run-on sentence if you like equals is the verb repeated several times and the way to read this is not to think of two X to the fourth minus 32 as equal to the last it is in this case but more importantly it's equal to the first and the first is then equal to this and the second is then equal to this and then the sentence ends over here that's the way you want to think about this now with equalities it doesn't matter so much if you equate the first in the last of course they're all equal but I'm going to show you here I have one period I'm going to show you another example now where it makes a big difference we haven't yet talked about solving inequalities we'll get to that but for the demonstration here I'm going to include one and you'll see why here's an example and just for the record here let's just assume that X is greater than one you'll also see why I did that okay three x squared minus three again I'm going to write a series of equal signs and other things watch what I do first I can see the three factors out so I have x squared minus one that's easy once again I see that I have a difference of two squares doesn't hurt to practice equals 3 times X minus 1 times X plus 1 now I'm going to put in an inequality for example this is greater than let me drop one of these okay I'll drop one of those 3 times X minus 1 now I know that what I've opt is X plus 1 X is bigger than 1 so whatever I dropped here is at least a factor of 2 perhaps bigger that makes this expression certainly bigger than this one and then finally this expression let me say multiply it out equals 3x minus 3 and then put a period ok once again one period and this is again one long sentence it's a mathematical sentence of course but it's just one sentence now here is the danger in reading this incorrectly if you read it correctly you say to yourself 3x squared minus 3 equals this which equals this which is greater than this which equals this so the net result is that the first here is greater than the last this is greater than the last that is the net result here however if you read it incorrectly if you assumed we just carried this down to the bottom we would have the 3x squared minus 3 equals 3x minus 3 and that isn't so so remember what this means when people write this and when I write this we are bringing these around this is simply shorthand instead of running off the page way off that direction in a very very long sentence we line the equal signs and the other signs the inequalities up so this is a point that's worth mentioning it's not often mentioned in a course like this but I think if you can get this straight it's going to make your work a lot easier okay well on that note let me mention one other thing while I'm in the in the pattern of making some by-the-way remarks let me make another one by the way okay here are some terms that I've mentioned just previously when I was talking about factoring and dividing polynomials I realized that I tossed in a couple of words that I haven't mentioned let me go ahead and mention them here we will see a lot more of them in the course but here they are ax plus B well you may have thought of that we have seen this as a 1 X plus a not using our polynomial notation this of course is a polynomial you can see that and it's a polynomial of degree one remember degree is the highest power of X and X to the 1 is here it is also called and this is the word I used earlier and realized I needed to say something about it it's called linear why because if you set y equal to ax plus B this is the equation of a line ok now we will study lines and that's line and linear of course we will study lines in unit following this but I wanted to mention that so you weren't confused by the terminology there are a lot of things in mathematics that have several words describing them and occasionally I may slip into one form or the other ax squared plus BX plus C is another one that should be fairly familiar you might have seen this now in our polynomial notation a sub 2 x squared plus a sub 1 X plus a sub 0 okay that would be the standard notation now this is a polynomial of course this is a polynomial of degree 2 ok degree 2 and it is called and this is strictly for historical reasons it's called quadratic ok because quad is something to do with a square and there is the squared power here that's the reason it's called a quadratic don't worry about that we will spend a lot of time later in the course talking about such expressions and then we'll set them equal to zero and talk about such equations but I wanted to mention these two because I had used the words and I wanted to clear up any difficulty they may be there ok well we're at the stage where we can sort of summarize some of the tips for solving equations that I wanted to pass on in this section so here's some tips ok first tip you want to make sure you notice and that's something we noticed in that that first equation that I looked at a moment ago the X cubed equal 25 X I said be careful that you don't divide by X because X could be 0 and that would not be allowed so the first thing you want to notice first notice and jot down okay where the sides of the equation that's the left and the right sides the sides of the equation are not defined okay so if I had divided by the X I would have caused an on definition to occur but there are other problems if I have 1 over X on the side of one of my equations the X can't be 0 because again division by zero there may be other problems usually I'll even give you the usual situation usually we're talking about division by 0 which is undefined remember or perhaps the even root of a negative number a little symbolism there there may be other things that go wrong but those are the two basic things you need to worry about even roots of negative numbers will lead us to what we call complex numbers which I'll talk about presently and division by zero course we've already said is undefined ok here's tip number two when you see the phrase show all steps or show all work which instructors like to put down what does that mean and why bother well what it means is leave a record it's what accountants sometimes call a paper trail ok you want to leave a record of your work so you can follow it so you and others not only your instructor but you have to consider that you at a later date can be considered another so you can follow your own reasoning mathematics is difficult enough without failing to put down the reasons for the steps that you've taken to solve a problem if you put them down then you'll be able to study them later that's one of the things you need to be able to do so you want to leave a record here okay leave a paper trail alright here's another tip here's some tips continued okay we get when you solve equations you get candidate solutions so check your candidate solutions now I call them candidates because what the algebra produces is not always a solution to the equation you started with that's something you have to be aware of check your candidate solutions for what kinds of things well possible human error we all make mistakes we'll add 2 and 3 and get 6 it's very easy to make a mistake like that so check for possible human error then you want to find want to check that possibly the only candidate solution ok the only candidate you have fails to solve the original equation now there's nothing wrong with that especially if you've done everything correctly what it means is so the original equation has no solution you've done all that algebra can do for you and what's left is the conclusion that there is no solution and that's a perfectly good collusion conclusion not every equation has a solution we saw that in an earlier example here's one more thing that can occur possibly when you have several solutions possibly some of them some of several candidates I'll keep using that word fail just as that one above failed perhaps some of them fail fail to work sometimes these are called extraneous solutions you'll see this in textbooks they're called extraneous solutions what it means is that algebraically you've done some operation that introduced solutions that do not solve the original equation very very common for that to happen and there's nothing wrong with it but you need to check your candidate solutions so that you can drop those out ok it's time for you to practice a few more problems and when I come back we'll talk about complex numbers you you

Channel: UMKC

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Keywords: Dividing polynomials, art of simplification, LCD, Zero product principal, Notation pattern, linear, quadratic.

Id: SPznX3uQvp0

Length: 86min 37sec (5197 seconds)

Published: Mon May 04 2009