well hello 3c and welcome to expanding binomials this isn't new this is still stuff from last year um but we'll complete this goal i can multiply binomials using the double distributive law and simplify more complicated binomial polynomial expressions so the double distributive law this is you won't find this word in the textbook this is something that i use most of the textbooks call it foil so multiplying binomials aka double distributive law aka foil most of the textbooks will refer to it as foil now when we have two sets of brackets being multiplied together we must use the distributive law twice in order to expand now you might be wondering why i have this thumb off to the side that's because i once worked at a school where they didn't call this either double distributive law or foil they called it the big thumb thing and the reason they called it the big thumb thing was uh because their teachers took a big thumb and this is back when they used chalkboards they had a big chalkboard and they had a big cardboard thumb and they would take the thumb and they would say okay we're going to start by putting the thumb over top of there and we're going to imagine that that term is not there at all and if that term was not there we're going to do it the same way as we did yesterday we're going to take this and multiply it through the brackets and when i take that and multiply it through the brackets i get 6 because 2 times 3 is 6 and then x times x gives us x squared so i get 6x squared and then when i do the next one here i go 2x minus times negative 2 is going to give me negative 4x and then we're going to take our big thumb and we're going to move it to this term over here and we're going to pretend that that term doesn't exist we'll put our thumb over that term and we're going to multiply this through as if it were just a question from yesterday and so i'm going to do 4 times 3x is going to give me plus 12x and then 4 times negative 2 gives me negative 8. and that's basically all there is to this except for the simplifying and remember once we're done expanding things we need to look to see if there are any like terms and there are i've got this one red term in this blue term here that come from multiplying the outside of the brackets and the inside of the brackets together and they are the same so i'm going to simplify by putting them together so the 6x squared doesn't go with anything a negative 4x and a positive 12x go together to make positive 8x and then we have that minus 8 on the end and now we are actually done as much as we can okay and that is what i call the double distributive law because when we moved our thumb around we just pretended that that term wasn't there and did the distributive law now sometimes this is referred to as foil most of the textbooks call it foil and i'll show you why i'm going to do the same question talking about foil but i'm going to change my colors here foil stands for first because i take the first term in each bracket and i multiply them together so there's where our f comes from for first so when i multiply the first two terms then i get that 6 x squared now i'm going to multiply the inside terms which should i spelled first wrong let's take the oops let's take that off there first okay i'm going to take my green pen here and we're going to multiply our outside terms and by outside i mean the outside of the brackets see these two things are inside those ones that i circled in green are outside and we're going to multiply those together those are the outside terms because they're on the outside of the brackets in other words i could take my big thumb thing and forget about these two inside terms there and when i multiply the outside terms i get negative remember got to watch that negative sign 2x times negative 2 gives me negative 4 x now i'm going to take my blue pen and i'm going to do the inside terms so the inside of the brackets here gives me 12x so i go plus 12x and then if i take my black pen i'm going to multiply the last term in each bracket see the last term in each bracket and i get oops i didn't want to take that away and i get negative 8. now it's exactly the same thing as i got up there it's just a different way of thinking about it so that we have the first the outside the inside and the last now one thing that i don't like about using foil is that it's not very useful when it comes to expanding out things that look like this stuff that have more than first outside inside and last because this one if i were going to multiply this out i could multiply the first and the outside and the inside and the last but i'm missing this thing all entirely and that needs to be taken into consideration so we can't just do first outside inside and last and here's where double distributive law comes in and i'm actually going to pull down my thumb and we'll use the thumb thing um let's actually bring that to the front so that it blocks something out okay so when we do this thing i'm going to start by putting my thumb over here and pretending that it doesn't exist and we're just going to multiply the x through the brackets so i'm going to take my pen and we're going to multiply the x there so x times 2x squared gives me 2 x and then it's going to be x cubed because i get 1x here and 2x is here so in total i have three x's being multiplied together now i'm going to take the x and i'm going to multiply it by that and i get x times 4x means that i have the four because this is just a one remember yesterday we learned about the one or the last lesson and x times x gives me x squared and then i'm going to multiply this there and that gives me negative x now i'm going to move my thumb and i'm going to pretend that that term doesn't exist and we're just going to multiply that 3 through the brackets so 3 times 2x squared is plus 6x squared and 3 times 4x is plus 12x and 3 times negative 1 is negative 3. now when we're done we're going to look for like terms so we'll have a look i've got an x cubed an x squared an x an x squared an x and just a number well the x cube doesn't go with anything so we're just going to leave down the 2 x cubed and we've got this negative 3 on the end that doesn't go with anything so i'm going to leave that negative 3 on the end that doesn't go with anything and other than that i've got 2 x squared terms and 2 just plain old x terms now those need to go together so i've got 4x squareds and 6x squareds so together that's going to give me plus 10x squareds 4 and 6 is 10. and now i've got a negative x squared and a positive 12x now this negative 1x is going to get rid of one of those positives leaving me with positive 11x and that's as simple as we can make it and so we're finished now we're going to try a few more examples of different types of things that you could see and you will see on the handout sheet that you're going to be doing so right here um this squared is on the bracket now notice this says expand each of the following simplify when necessary so sometimes it might not be too necessary to simplify most of the time there'll be a simplification now what i what this two out here means is that i actually have two of these sets of brackets multiplied together remember when i have x squared that means that i have an x multiplied with another x well now i have a bracket squared so that means that i have two of these brackets multiply together and if i have two of those brackets multiplied together then i can just use the double distributive law so i'm going to take this 2x and multiply it through that's going to give me 2x times 2x is 4x squared and 2x times 3 is going to give me plus 6x now i'm going to take this 3 and multiply it through the brackets which is going to give me a plus 6x again and a plus 9. and now these two terms in the middle are going to go together and we get 4x squared and then a 6x and 6x is plus 12x plus 9. and that's as simple as that one now the next one this one looks a little more complicated because i got this number out front but here's what i want you to do i want you to um let's say take that thummy thing ta-dah now there's our thummy thing we're going to take this thummy thing and we're going to put it over the two and we're going to pretend that it doesn't exist for a minute um just for a minute though because we don't want to forget about it entirely so here's what i'm going to do i'm going to pretend it doesn't exist but i'm going to put my answer in brackets here and i'm just going to multiply this out so we're going to multiply the 4x by the 5x and get 20 x squared because 4 times 5 is 20 and then an x times an x is an x to the exponent 2 and then 4x times 2 is plus 8x and now we're going to multiply this one through negative 1 times 5x is negative 5x and negative 1 times negative 2 watch those negatives a negative times a negative gives me a positive 2. now that number out front was actually there so we can't forget about it but i'm going to give one more step before we take it into consideration or well let's put it down there we know it was a 2. we know it's there that thumb's not fooling anyone it is there okay so we've got the 2. out front but i'm going to simplify what's in the brackets this is going to make it just a little bit easier 20x squared and then we've got positive 8 and negative 5. positive 8 negative 5 is positive 3x remember these five negatives are going to wipe out five of those positives and i'm left with three x's plus two now that i'm done i'm going to pay attention to that two that we had put the thumb over top of and i'm just gonna multiply that through the brackets with normal distributive law because there's only one thing so i just have to double each term 40x squared plus 6x plus 4. next and last and this is a bit of a doozy okay that's a really big problem the first thing we have to do is remember what this squared out front means well that squared out front means that we're going to have two of those brackets together so three x minus two three x minus two and then subtract two x plus one and three x minus seven now this minus is really important in here subtraction changes things it changes signs after the brackets remember when you have when you're subtracting a negative it changes to a positive well stuff like that's going to happen down here too so we're going to take that into consideration the first thing we're going to do though is forget about this 2. we're going to put it out front and we're going to forget about it until we finished expanding this so we're going to multiply the 3x through so 3x times 3x is 9x squared 3x times negative 2 is negative 6x negative 2 times 3x is negative 6x and negative 2 times negative 2 is positive 4. now i'm going to leave this negative here and i'm going to put a set of brackets and i'm going to multiply through this bracket so 2 times 3x is 6x squared and 2x times negative 7 is negative 14x and 1 times 3x is 3x and 1 times negative 7 is negative 7. now here's what i'm going to do i'm going to simplify in both sets of brackets here so basically i'm just doing two different problems at once i could go all the way down with one and all the way down with the other if i wanted to but i prefer to work all the way across so i've got 9x squared now negative 6 and negative 6 gives me negative 12x and plus 4. and in this brackets i've got 6x squared and the negative 14 and the positive 3 go together to give me negative 11x minus 7. now i'm going to put this 2 through the brackets same as i did up here after i expanded and simplified then i put the number through out front and i'm going to pretend like there's a negative 1 out front here and i'm going to put it through the brackets that it's in front of so this is what we're going to end up getting it's going to say 18 x squared i'm just going to double all of these things in here minus 24x plus 8 just doubled everything just multiplied by 2. and now when i change when i multiply through by negative 1 the only thing that changes are the signs so negative one times six x squared is negative six x squared negative one times negative eleven is positive eleven x and negative one times negative seven is positive seven now the very last thing i have to do is collect like terms and so i get 18x minus 6x is 12x squared okay the squares now notice it's a squared it's not an x to the fourth or anything like that when you're collecting like terms the exponents stay the same i get negative 24 and positive 11 is going to give me negative 13x and then the last thing we're going to do here is take positive 8 and positive 7 and get positive 15 and that is the end of this lesson