 What's up everyone welcome to tutor terrific. I am so glad to be here in my first live stream I don't know who's gonna show up. I had issues. I use StreamYard as you can tell. Oh gotta get used to that right up there if this goes well, I'm going to definitely look at Purchasing something from them. This is their free version right now like test it out, but Welcome everybody. I had some issues Posting these live streams on the channel so that you can see them. It wasn't showing up in the videos feed I don't know why when I scheduled this thing, but I made subsections under that So that you can see Upcoming and past live streams and when I'm live too, so those will show up soon But hey, this is my first time doing this and I am super stoked That YouTube makes this possible. I'm sure you've heard that before but I'm new to the game But I want to help people. I want to help people all over the world. I want to help people No matter where they are with what they need help with when it comes to Educational endeavors so let me know either in the Chat over there or in using that email address as I put in the chat. Let me know What you need help with whether it's something content related or a specific problem. You're just having trouble with a concept I'm here to help and I have something planned just in case No one is a game right now and that's just fine. I've got a lesson on partial fraction decomposition Which people seem to be Think is quite hard and that's just fine. So We're gonna do that until people show up And we'll see people show up. It'd be awesome if they did but Yeah, if I get a lot of people when I get a lot of people I will Certainly any donations any Things you do like super chat or the little thanks heart are awesome. I will definitely be Prioritizing the super chats that come in and the questions that people have if they use super chats So we'll see lots of things to test out here, including stream yard itself very very stoked for this whole thing So without further ado, let me get my screen shared here and let's look at What I'm doing on This particular Thing here called my screen Just Let me figure out got to share that screen You know how to show my screen. I thought this was it. Yes There it is We're gonna share this one over here All right, so you guys should be seeing that now Partial fraction decomposition by the way, I have discovered a long time ago one of my students showed me this thing called Miro and Yes, of course we've had zoom for you know, two three years now and It's been really annoying and their whiteboards aren't very great But my friends showed me this thing called Miro and it blows Zooms whiteboards even their new whiteboard feature out of the water This is a game changer and it's free and you can share you or basically your whiteboard universe space Which I will show you here. I have this lesson planned for partial fraction decomposition, but I'm not even Like I can go anywhere I want and I can add graphics I can add graphs I can add images I can add YouTube videos I can actually embed videos into this This like universe whiteboard space and multiple people can be Live going on here and working on it, which is great At the same time and you can like if you have multiple people in here You can have someone up at the top here You'll see all their icons and you can click on their icon and go right to where they are if you're lost and follow them around while they're writing it's really cool and it doesn't seem to Limit who how many people you share your board with they call it? This is my board called YouTube lives But I have another board that fits every single one of my two drink students every single one They get their own little section of the board and they just know to go right to it when they start my private students sessions so Yeah, let's do this partial fraction decomposition. Okay, partial fraction decomposition is You could consider when you're getting new to this topic the opposite of Combining fractions with unlike denominators or like denominators, but combining fractions with Rational expressions in them. So not just numerical fractions, which are easier but algebraic Expressions in the numerator and or denominator like we have here so This understanding this guy's is going to require you to have some algebra background Certainly if you're not using common core style math, you should be through algebra one for sure definitely But having algebra two under your belt is a plus being in pre-calculus is where you would learn something like this Maybe some of your schools call that trigonometry if you aren't in If you're in a common core setting, I would want you to be in math three before you start to learn this but hey If you think you can follow along and you've combined fractions before you probably keep up But let me tell you It's the opposite of what I'm about to do, but I want to give you an example just to get you in the zone here So if we have two over five plus x plus six y over six minus x we have different denominators And so we have to combine them the proper way. We have to combine them Using Least common denominator we can use a other common denominator, but the least common denominator will assure us that we don't have to do any Simplification later. So now we need to find the LCD. Okay, that's the first step so when you have Rational expressions in your denominator like this the least common denominator will have one copy of Every unique factor and one copy of every common factor Again, let me say that one copy of everything unique one copy of everything common Okay, so there's lots of different ways that teachers teach that but this is how I do it Let's look at the denominators here We've got five plus x and six minus x you might say well They both have an x but x is not a factor Five plus x is a factor and six minus x is a factor and that's all I care about are the factors So these are unique. They are not identical. And so the least common denominator will have one copy of each in its Product, okay, these common denominators a product of factors Okay, so this right here is the least common denominator, okay, and you can choose to multiply that through With the double rainbow method or the box method whichever like you like better you would get 30 Minus 5x plus 6x which is just x Minus x squared and I'm a stickler for writing things in standard form. So you'd have you'd switch the order here and put the negative x squared first the next lowest power of x in the middle and then the constant term at the end plus 30 and Then you'd have your least common denominator all multiplied out or you can just leave it factored like we have up here It's up to you. I like it. It looks a little cleaner like this So I'm that's my preference. So what we're doing is we are going to multiply each of these fractions by what their denominators were missing from the LCD, okay? That's what we're gonna do right now. So This first one 2 over 5 plus x it was missing the 6 minus x From the LCD it didn't have it downstairs. So I need to multiply using the denominator numerator same quantity rule Okay, let's find what that is. I can multiply a fraction by Any non-zero quantity as long as I multiply the numerator and denominator by the same non-zero quantity. I Will get a fraction that has the same value. That's really helpful because I Can change what the fraction looks like even without changing its value doing it this way Zero doesn't count. You cannot do this with zero at all at all at all when you multiply by zero over zero first you're multiplying by something that's indeterminate doesn't have a determined value and You're getting rid of a lot of information when you multiply something by zero because the product is zero and there's nothing left So we don't want to do that What we do want to do This we want to multiply by the factor again From the LCD that was not present or all of them that were not present in your Fractions denominator in that case it would just be six minus x so two over five plus x turns into this Now the we'll do the multiplication in a minute, but next what we have to do is we have to Look at the next fraction six y over six minus x and say hmm. What's missing from here? Well, I'll show you Six y over six minus x. I care about the denominator guys six minus x downstairs That's missing from the LCD the five plus x And so I need to multiply the top and bottom by five plus x Okay Don't start cancelling stuff if I cancel like this and like this because I have my Cancel trigger my cancel finger on the trigger that's not gonna be good. So don't do that. Okay, you need to put You need to multiply okay, don't put anything you need to multiply so on top we're going to distribute the two in the first fraction That's 12 minus 2x and Then other fraction that'll be six y times five that's 30 y Plus six y x or six x y if you like doing things in alphabetical order like me and I'm going to use that nice Multiplied out version of the denominator negative x squared Plus x plus 30. It's not just any denominator. It's the least common And so here we have now Two fractions that have the same denominator, which means we can add them How do we add fractions with the same denominator? Well, we make really one fraction with that like denominator Equal denominator. I should say so one copy of that Negative x squared plus x plus 30 for one fraction now Just one fraction on top We're gonna add these two expressions now. We add using the rules of like terms Terms that are like Can be added. What do they have to have they have to have the same variables to the same exponents? If they don't you can't add them Any more than you could just write them next to each other with a plus or minus sign between them So let's look here 12 minus 2x 30 y 6 x y None of these are like that's okay. It doesn't mean anything's wrong. It just means you have to write them With a plus or minus sign between all of them You just end up with this Right here Fortunately, that's really weird complicated But most of the reason why is because I had a variable up top in the beginning So that's combining fractions with unlike denominators. Okay? We're gonna do the opposite of that. We're gonna start with something. Thankfully not something this crazy over here But we're gonna start with the end and go back to the beginning Okay Step one when we do this. Okay, there's a specific step order when you have your fraction If it is improper you must divide so you must divide if Improper, okay, you must divide if improper So if you have a fraction with a higher degree By the way, I'm rolling into do examples that have one type of variable in them Like x or y and it's the only one but If it's improper meaning the upper degree is higher in the numerator than the lower degree That means the higher power of x is on top Then you have to use long division or if you can based on the format synthetic division To get a quotient and a remainder and that remainder as you know from long division You're gonna write that remainder over the divisor over the denominator That will be proper and then you can move on. Okay So if it is improper Right remainder, I'll just call it r of x because it's a polynomial itself right r of x over The divisor I will call d of x and continue To step two, okay Now step two. What is step two? Step two is to set up Your denominator properly you need to factor the denominator fully fully factor Denominator you need to look at what your simplest factors look like. All right, you know, I'm a demon Denom for a denominator for short. Okay? You want Factors that look like this. Okay? one of these two types Linear factors Those we're gonna say have the form px plus q So x is our variable P and q are numerals coefficients and It can be raised to any power Obviously the more complex it is the more the higher the power of m is or they could be quadratic a Little bit more involved if they're quadratic, but those could look like this ax squared plus bx Plus c notice how x is squared in these and that could be raised to any power We'll just say different one than the other one to the end now these are Irreducible these quadratic ones. I'll explain what that word means irreducible when something's irreducible over the reals Or over The real number set are that means I can't factor it. Okay into nice rational numbers Okay, or even irrational numbers This thing has complex solutions if it has any so that's what we want We want an inner reducible over the reals. No factoring possible. That's the main point Okay, if it if you do have a quadratic factor that you can factor further do it and turn it into a linear factor, okay now Step three is really depending on what type of factors you have for linear factors of that form px plus q to the m We have to set up Partial fractions. Okay on top the numerator will have just a number Each numerator will have a unique number Well, they could be the same. We're just going to pretend like they're not give them each a different subscript and a 1 a 2 a all the way through up to the power itself and That the linear factor is raised to on bottom We will have the original linear factor But each time we write it We write it to one higher power Wow pq plus q come on. Hello px plus q Now it'll be written to the squared power It's the second term we go all the way ppx plus qq px puts p to the fourth all the way up to whatever m is px plus q To the m. Okay, sometimes you're really gosh. All these are this long. No, they're not They're not all this long. Do not worry. Okay next For the quadratic factors the setups a little different on top will actually be linear B1 x plus c1 For the first one and on bottom you'll have that quadratic factor Raised only to the first power remember. This is irreducible Okay, the next one will be a whole another set of coefficients with that x be a b2x plus c2 Just the little two subscripts mean it's different than the other ones in the previous fraction and that will be raised That'll be divided by a denominator that is raised to the second power So we're going to continue doing this All the way up until we have m fractions matching the power. Oops. I'm so sorry. That should be an n It could be different than the other ones. We're going to go all the way to n whatever it is be nx plus cn all over A x squared plus bx plus c to the can you guess it? and Okay, again looks super complicated, but What we're going to do next Will hopefully make sense of this okay getting these coefficients on top or these numerals on top is the work that you must do And so we need to do that So here's our first example Okay x plus seven over x squared minus x plus six now When we do this We have to start by making sure we have a proper fraction and it comes to degree and We do here the top degree is Smaller than the bottom degree. How do I know degree? It's the largest power of x in the polynomial that you see So the numerator polynomial It's x to the first power the denominator polynomial The degree is to the highest power of x is to so the bottom the denominator has a higher degree So it is a proper fraction. So we're going to skip right to factoring the denominator Okay, look here. This is a trinomial a quadratic one and so we look and see that the first coefficient is one So we've got a simple method for factoring this what two numbers multiply to negative six the constant term and Also add to negative one the coefficient of the middle term and the answer I'll give you a second to think about it Six is one times six or two times three Negative six we've got to have one of those be negative Combination that works such that they also add to negative one would be negative three and positive two like that And so what we do thankfully get to skip all of the intermediary group factoring steps From the AC method. We just skip all that we get to write Two parentheses already multiplied together each start with the x and Then you add in these found numbers From your little exercise over there Okay, hopefully you've all seen this before if you're not familiar with that you're gonna kind of be lost Okay, so maybe you're not done with algebra one or you aren't done with math one yet ouch. You've got to know your factoring Okay, that's okay. I shouldn't say ouch as you just say This is a little bit beyond what you're able to understand most likely But that's factoring and I fully factored the denominator. I have beautiful Factors that are linear, okay Now but they're they they they're each raised to the first power. Okay, so I must have One Partial fraction for each one of these, okay The first one will just be a on top over x minus three It wasn't raised to a higher power So I don't have to add more fractions and more letters more versions of a on top. I'm done with that one But I've got another fraction Because I've got another linear factor on bottom x minus two to the first power So I'm gonna have to add another fraction. That's got x plus. Sorry. I keep saying minus two It's plus two on bottom on top, I'm gonna use a different letter for my numbering scheme because I'm starting over with another linear excuse me factor Okay, and I says as I said before We're done creating these partial fractions We have to find a and b. Okay, that's the goal right now is to find a and b so Here's what we're gonna do We are going to multiply by The denominator of the original fraction In factored form might be a little easier Imagine doing that from here to here imagine multiplying both sides by x minus three Times x plus two What's gonna happen? Well, I'll tell you what's gonna happen. What's gonna happen is that the Entire fraction the entire equation is gonna lose all of its denominators It's very easy to see that that happens on the left side because you're multiplying something That's already divided by x minus three times x plus two by x minus three times x plus two So this is gonna delete this and we're just gonna get x plus seven over here, and that's what we want on the other side If we Distribute this in we can see that part of it's gonna cancel so a Times x minus three times x plus two divided by x minus three the x minus three is gonna cancel And you're gonna be left with a times x plus two on top Okay, the other one Multiplying x minus three times x plus two the x plus two is gonna cancel Which is this leaving you with b? x minus three Okay And that's a lot of work. We've got more to do. We've got to do the following we have To distribute these a's and b's to these linear factors that now multiply by them Then we're gonna do a little investigative work with this equation because we've only got one equation We've got two unknowns to solve for So watch this a x plus two a oh Oh Get ahead of myself there a x plus two a plus bx minus 3b like so and What we're gonna do now is we're going to Combine like terms Meaning terms with the same variables. Okay. There are two terms With an x in them on the right and So we are going to combine them like this and factor out that x So there's an ax and a bx That's a plus b times x Then we have plus two a minus 3b not really common at all We'll just stack those in the end. Those are gonna make up a constant. Okay. They're just numbers Now if I'm saying this is equal to x plus seven, I'm actually Implying quite a bit. Okay. What's the coefficient of the x term on the left? It's one This implies because I have only one x term on the right side That a plus b has to equal one This is the first equation that I've created where I can actually solve for a and b But that's not enough because I need to and I only have one. Okay So what we need to do now is We need to set the other portion of This to seven. Okay to a minus 3b Equals seven. Okay. Now. I have the proper number of equations To get a and b to So each time you do a partial fraction decomposition the more complicated what you started with is the more equations You're gonna have but you're gonna be able to solve them as a system like this Sometimes you get some very easily in this case. We're gonna work for it I've got this beautiful system of equations here that's set up just perfectly for elimination as my means of solving And so I'm going to go ahead and multiply the top equation by two s by three Which will give me and I'll write it below 3a plus 3b Equals three like this and I add these two equations the b terms cancel and I get 5a Equals 10 it's 2 a plus 3a is 5a 7 plus 3 is 10 We divide 5 you can see a Equals 2 we found one of those numerators awesome Finding the other one is a cinch you back substitute Take that value got for a and plug it into any of the equations you have here Top one is the easiest imagine it without being multiplied by three You've got a Plus b equals one well now you can substitute in what a is it's two plus b Equals one So if I subtract two then I'll get b b has to equal negative one Okay So I found a and b now you're not done what you got to do is go back up here You got to write The fractions those partial fractions with their found numerators. So a was two and b Was negative one so you could write that instead of plus a negative one you could write minus one over x plus two Now what you want to do obviously is you want to make sure that this is right, okay How can you make sure this is right? Well, you can combine the fractions do the exact reverse Using older maths that you learned years ago. Okay These denominators aren't like so I have to multiply by the lcd We've worked with the lcd enough in the example that you know the lcd is just the combination of these two factors together So the top the first fraction needs the x plus two Because it didn't have it in the lcd The other one needs the x minus three multiplied by it because it didn't have it in the lcd In the lcd. I mean it didn't show it in its denominator That part of the lcd was missing So when you multiply through of course The denominator is x minus three times x plus two. I'm just for time's sake. I'm going to leave it un factored I mean Factor not multiplied out the top is what I care about Two into x plus two. That's two x plus four Now be careful here. This is a minus sign. I have to distribute it To all the terms that follow it. So one times x times a negative is minus x Uh One times negative three times the negative is plus three Now here in this case. We actually do have some like terms Two x minus x is x Four plus three is seven Multiplying the bottom get x squared minus x minus six Which is what you started with so it checks out that this right here Two over x minus three Plot minus one over x plus two is correct Now you're thinking man, I this is crazy. This is so hard. This is so much work I could just guess And figure it out. No, you can't Yes, I can I can factor it and then just split them up and then guess what a and b are No, you can't these are not intuitive numbers It's because of the work required to get to them that you can't just intuitively just Guess what they are you can't okay? So that is not a good strategy A good strategy is this strategy and it's up to you to get used to it so that you're not Lost okay, but takes practice so we're gonna do another one Whoa, look at this guy. This guy is a little bit more complicated. Oh my goodness So if this guy is this complicated Is it a totally different method than before? Hi, I see you. Thanks for joining Ask for any help you need on anything. I'll stop what I'm doing and I'll help you Or bring your friends in I'm here till 11 Pacific standard time so What I want to do Now is step one which again is look here. Is the degree of the numerator Correct Is it smaller than the denominator? Yes, I've got degree two on top with this two here Or degree three on bottom And that means it's a proper fraction again So I don't have to do any long division which would really oof that would be a pain here okay so What we're gonna do is we're going To Back to the denominator. Okay We'll just keep writing the numerator. Don't need to change it the bottom We can factor start by looking at all the terms. Is there something common to every term? We could pull out the greatest common factor Okay, and that's x every term has at least an x in it so that means that x is the greatest common factor I'll factor that out first I have x squared plus 2x plus 1 left Okay Is x squared plus 2x plus 1 irreducible No, it is reducible. Okay, you might not see it But this is a perfect square and now there's ways to tell for example Is the coefficient in the middle Double the final coefficient squared. Yes, that's one way to do it, but you don't have to do it that way you can do this The x method Can I think of two numbers that multiply to one and add to positive two? Yes, of course Can you If you can Good for you. It's one in one Okay Good for you. All right. So that means I can factor the denominator further Which is good because quadratic factors are more work The algebra in this algorithm above is more complicated But doable just more work. Okay So I've got that x there and now instead of writing just x squared plus 2x plus 1. I'm going to write two parentheses Each has an x and each has a plus one from our little Investigation there Okay, so really what I have Are three linear factors, but not really they're the same like I said, it was a perfect square So I shouldn't write it like this I should write it like this Okay, so what I have Is a repeated linear factor whereas before in the previous example I had distinct linear factors Not a problem not a problem at all. Okay So Here's what I'm going to do. I'm going to set this up Now I can think of x as x plus zero. Okay So I'm going to have an a over that x plus zero And I don't need any more copies because it was just the first power Then it's next one I have a linear factor, but I have to to the second power of it So I'm going to have a b1 Or you could write c b and c it actually works. It's a little easier if you do it that way um b over x plus one and then c Over x plus one squared. Why because it was squared here I have to repeatedly write these factors as denominators raising the power until I get to the power of the fully factored denominator original And so I have this here It's now ready To go through and multiply everything by the least common denominator Well, sorry the denominator as it's factored. So I'm going to multiply Everything by x times x plus one squared on the left and the right This is going to cancel over here Only parts of it are going to cancel. So the left side 5x squared plus 20x plus six Set in stone the other side The first fraction a over x the x is going to cancel Leaving us with just a times everything it already didn't have You saw that pattern already and then b One power of x plus one is going to cancel So that b will be multiplied by x times and the other x plus one We'll distribute that then Then c c the x plus one squared will cancel. So we just multiply that by x Okay, a little more work here Now I have to do some foiling and I have to do some multiplication so I can start to Group stuff differently in a way that's helpful like I did in the previous problem So foil here And multiply in by a that's ax squared Plus two ax Plus a for one squared then Bx times x bx squared plus bx Okay, and then cx Okay, like I said before group all like powers of x Okay Don't do anything to the left side yet So and factor out The x squared from them So I've got Two terms with an x squared a and b So that's going to make a plus b x squared after I factor the x squared out Then we've got first power of x terms I've got a two a Plus b plus c x All right, cool And I only have one term without any x's. It's just a oh, that's nice. Okay, you'll see how awesome this is But first you're like, oh my gosh, I have three different letters to solve for I'm gonna have to have three Variables. Oh, no. Yes, you are but that's okay Trust me. This isn't nearly as bad as it looks and you're not going to use elimination to solve it Okay So If the left side equals the right side, we could set coefficients equal to each other just like in the previous problem. Okay What we're going to do at the front is we're going to set five equal to a plus b Five x squared has to equal a plus b x squared Because the power of x the next one's the hard one 20 Equals two a plus b Plus c Oh, dear three variables in that equation The next one Is the simple one the constant terms have to be set equals six equals A. Oh my gosh. I already know what a is a is six Oh, that's so nice Okay So what would I do next if I know that a is six? How would what would I do next I would Find b because the first equation has just a plus b So I will write five equals six plus b Plugging in a And I don't know if you can see that But six plus b equals something less than six five. That means b has to be negative one awesome Okay, and then this isn't so bad now to find c. We just plug a and b into the second equation 20 equals two times six Plus negative one Plus c So 12 minus one. That's 11 20 equals 11 plus c I think you can see right now just with mental math c equals nine Okay All right, we know what a b and c are so we come back To our partial fraction decomposition And we oh, I just realized I scared away that one person Yikes, it's kind of awkward when you're that one person watching someone else's live stream on youtube So if you do that again, sir, I will make sure not to Say hey, buddy. I see you. Hey ask questions. I won't do that to you. That's kind of freaky All right, so come back up here Our partial fraction decomposition Is going to be the following it is going to be six over x And then we've got a negative one for b so minus one over x plus one And then plus c which is nine Over x plus one squared Whoa, that parentheses out of control What is going on there? There we go. I am going to leave it as an exercise to the listener one reader to check this but I can assure you And it's not going to be nearly as easy as the previous one to combine those but you can't do it all right there is my lesson our partial fraction decomposition And it's been about 44 minutes So at this point I would start to wrap up if I'm not getting any questions anymore Just get to a last kind of shout out for any questions we have And um, then I would answer those of course again giving Prefits to the people paying the super chatters So I'm hoping this goes big. I'm hoping people see the use in it I'm going to be able to do this a lot and get a lot of Good usage out of it for people so they can get their questions answered Not just in math, but in sciences writing I'm here to help all around and I'm not going to sit here and say I have all the answers because I don't I might be wrong at times and your feet are correct. Just be respectful. So I don't have to ban you Or move you from the chat because you're being a troll or you're being a jerk or you're being a degenerate Don't do any of that stuff. Um, be kind be professional and let's let's get some let's get some stuff answered. That's why I'm here and This is going to be a really I think a really good thing for youtube If more and more people start doing this, but I'm going to start it. All right And you know, I'm just being facetious. I could be literally the 1500th person to actually do this. We'll see. I just haven't come across it And um, I'm probably the only one doing it with Miro. So we'll we'll see about that Well, like I said guys, um, I would wrap up at this point If I don't have anybody asking any more questions And it's been a pleasure for all of you who are going to see this post And for you one person who I scared off. I love you. I don't hate you. It's just awkward. I get it um, but yeah, guys, I'm Here to help I love helping if I could support my family doing this. I'm certainly gonna try and um Yeah, I love teaching so Teach more people. We'll see what happens. All right guys. Thank you so much for watching It's been Falconator signing out