 And welcome back today. We're talking about multiplying polynomials. I was doing more more complicated examples I know in my first video. I stated that I was gonna do three videos. I'm actually gonna cut that I'm just gonna do two videos The second one that I'm gonna do is gonna encompass the two harder types of multiplying polynomials I just have one example of this one, which is a binomial times a trinomial And then I have another one where actually we're gonna have three different parentheses Not sure how to expand that here in a moment. Okay, um in this example I'm just gonna kind of go over the basics of this and I'm just gonna do one example Now what you got to think here is this is a process you've actually have seen before if you're at this level of mathematics You've also seen this something like this do a quick example x minus five times the quantity x plus two This right here. It's a binomial times a binomial. You've done this before. It's called foiling First outer inner last. That's why we call it foil. It's a it's an easy easy way for us To remember what to multiply so you take first x times x to get x squared you take your outers x times 2 to get 2x Inners I negative 5 times x here to get negative 5x and then L is last negative 5 times 2 for a negative 10 These two middle terms here are alike. So I'm gonna add those together for a negative 3x minus 10 Okay, now that that right there. That's foiling. This is something that you have done before now This is actually the exact same process of what I'm about to do over here except for foiling is a little bit easier This is a little bit harder because we just have more to multiply. Okay, so that's kind of some backstory there Alrighty get rid of that and let's get on to our actual example here So what I'm gonna do is I'm gonna take this example And I'm gonna show you the different multiplications that I'm gonna do okay now first What I suggest here is that you always take the smaller one times the larger one meaning I'm gonna take this Turn I'm gonna take these two terms times the three terms over here, and I'm gonna start with this a here I'm gonna take this a times everything in here first I wouldn't I wouldn't switch that around if you flip that around what would basically happen is you take Two times everything here 5a negative 5a times everything over here twice and then it's just it's just a messy process It's actually easier to go the other way. So I'm gonna take this a Times two first okay, so that's gonna be to a I would take a times negative five a negative five a square The only thing that's really increasing is the variable There's only an a that we're multiplying by here So those that's just the variables are increasing so that's gonna happen over here is a times a square is gonna be a to the third Okay, so there's your first multiplication Now different teachers will call this different things some call them rainbow some column arcs I don't usually use them very often But I know it's helpful for students to do this so they can keep track of what multiplications they're doing So I'll show it here, but anyway the second multiplication that I'm gonna do is I'm gonna take this negative three here And I'm gonna multiply it times everything inside. So take negative three times two to get negative six Negative three times negative five. We be careful with the double negatives there So this could be a positive 15 a negative three times negative five a and then last is Going to be negative three times a squared, which is just a negative three a squared pretty easy there And now I have this color coded so that you can see okay So that I got the top three multiplications to get three here the bottom three Multiplications here to get these three red ones over here. So basically you have a binomial times a trinomial You have a two term times a three term notice two times three is going to give you six different terms Okay, now that's actually going to happen every time you could figure out how many terms You're supposed to have based on what you're multiplying by two terms times three terms is going to give you six terms Okay, so it's just one strategy you can use to kind of help yourself out to make sure I'll do I have all the multiplication supposed to have well two times three. Yes. Yes, I do okay Anyway, now we need to add some like terms right here is a to the third negative a to the third is going to be negative two a To the third and go down one step to your a squared, which is right here Oh, hey, what am I doing? What was going on? That's a to the third. That's a squared That's not even close to being the same. What are you doing? Okay? Backtrack a to the third is all by its lonesome all by its lonesome. There's an a squared So negative five a squared negative three a squared is a negative eight a squared very handy to make sure that you Label these like I did with the underlines because it was pretty easy for me to realize my mistake right there But anyway moving on I got the got the cubes. I got the quadratics. I got my squareds there And I got my a's left over one two three lines for that one one two three lines for that one that makes a 17 a and then minus six there we go. All righty Yeah, that's it. So that was multiplying polynomials. That's one of the different problems that you will see Again pretty straightforward The difficulty here is that you just have to keep everything organized Just make sure like with these underlinings that I do with these arcs and that kind of things find something that works for you So you but kind of keep track of everything Anyway, the next one I'm going to do Expanding a power of a binomial. So this one This one is always trip trips everybody up There's a really really common mistake that happens with these type of problems I'm gonna show you the mistake first and then I'll go through and actually work out the problem Okay, now here's the mistake a lot of kids like to do this They like to say oh well I just take this this this three here this exponent of three and I'm just gonna apply it to the k and apply it to the Negative five that is actually incorrect. That is way way off. You're forgetting a lot of multiplication that you have to do So this is not is not K to the third minus five to the third it this this is not even close to what what the actual answer is going To be okay, that's the most common mistake that I see so just don't don't make that don't assume that okay So this is actually what we're supposed to do. Okay, we have k minus five this quantity. We have three of them So that's what I'm gonna do. I'm gonna rewrite this as k minus five k minus five and K minus five get you have three of those parentheses. So this is what we mean by Expanding it. Okay, you're gonna write it out like this now before I go any further There is a way to to quickly do this there's a pattern that's associated with higher powers of binomials You can use Pascal's triangle To to kind of get through this quickly, but I don't have time to go over Pascal's triangle and just kind of showing you the basics of multiplication If you if you can figure out Pascal's triangle on your own fantastic all the more power to you But we're just gonna stick with the basics here. Okay, so now back to back to the problem So I have k minus five times itself three times what I'm gonna do is I'm gonna multiply now You could only multiply two of them at a time don't you can't multiply all three at once That's just way messy. All right, so what I'm gonna do is I'm actually gonna take these last two over here and multiply them So this k minus five over here. I'm just gonna leave leave this guy alone I'm gonna multiply these two right here now. It's a binomial times a binomial. So this is just foiling So that the process I just showed you a moment ago what you're already supposed to know That's that's it. That's what we're doing here. So this is going to be k squared minus 10k Plus 25. Okay, you're for your your first outer inner last is going to multiply to this now You might want to pause the video and actually multiply it out if you don't know where this comes from You might want to multiply this out yourself to get that I skipped a few steps there But anyway now moving on kind of the same deal what I'm gonna do is I'm gonna take Just like the last problem. I'm gonna take k times everything first and then negative five times everything now I'm not gonna show all my steps here. I'm not gonna do the little arcs and that kind of stuff I'm just gonna kind of truck through this. Okay, so I'm gonna take k times everything. So that's k to the third k squared and 25k there we go Basically that first multiplication is just taking k times everything all of these are their k's are gonna increase by one So to the third squared and then this 25 is actually gonna get a k. All right So that was kind of easy multiplication next is I'm gonna multiply times negative five So negative five times k squared is negative five K squared nice looking five there get rid of that. That's Atrocious there we go and negative five times negative 10s a positive 50k Make sure you keep track of the negatives and then negative five times 25 is Five quarters is a dollar 25. So that's negative 125. There we go. All right and Add some like terms combine your like terms together K to the third is all by his lonesome Here's my squared. There's a squared. So I have negative 15 K squared add the coefficients together, right? Now go to the k's double underline here double underline here. I get 75 of Those k's remember the coefficients increase not the exponents Okay, so the numbers out front are increasing but not the exponents themselves the exponents are staying the same Where am I this is why we underline things? Oh negative 25. That's where I am Negative 125 excuse me. All right, and that's it. That's how you expand a power of a binomial not there's higher there's higher powers you can do to the fourth or to the fifth or to the sixth and That's actually where Pascal's triangle might be easier to use is if you take K minus five if you take this to the seventh eighth or twelfth power or something like that It's going to take you a very long time to multiply and Pascal's triangle that method. It would actually be a lot quicker But anyway, that's that's that's something else for another time Again, make sure that you develop a strategy to this and just keep track of everything whether it be showing the arcs on your multiplications, I still do these underlines just to just to make sure I'm Just to make sure I get everything as you see my mistake earlier that underlining actually saved me on that my mistake But anyway, I think that's about it. That's multiplying some polynomials some little bit more difficult examples and Thank you for watching. See you next time