if a blast off it's pretty cool um last week we started our rational expressions rational expressions just means the fraction unit we had three steps to every question that we talked about the first step every single time is factor and some questions you can't factor so if we look at this first multiplying of rational expressions question there is no factoring that can be done there's no factoring because there's no addition or subtraction so that's like the giveaway that there's no factoring that's available the second step to any question is that we need to state our non-permissive values which we called NPD's just a reminder of what NP B's are there are things that do what yeah their values that cause us to divide by zero they you're not allowed to buy buy zero in math so a non personal value is something that would cause you to divide by zero if you look at this question the two areas that I would need to check for dividing by zero would be right here and right here cuz those are two different spots or I'm doing division the blue one that I'm highlighted is divided by ten there's no letter there so there's no way I could make ten equals zero the red one that I've highlighted is two times C there is a possibility that I could make that equals zero what value of C would make that zero close that's one of the people fall for it because usually it's like two plus C or a in the people are like oh it's the opposite number but when they're side-by-side like that and there's nothing in between them the way you would make it zero as if it was two times zero so what value of secant you have you can't have zero right so anytime there's a monomial letter that Leonard cannot equal zero awesome the third step that we did in every single question was we simplified simplifying when it's multiplication of rational expressions well we like to do is combine it into one rational expression so one fraction and then simplify anything that can simplify the way we combine them into one is we multiply across the top and we multiply across the bottom and that becomes one fraction overall on the top C squared times 5d we say numbers go with numbers and letters go with letters when it comes to monomials five there's no other number on the top four multiplied by C squared there's no other C's two x and D there's no other D's two x so the top just becomes five C squared / on the bottom we have 10 times to see they're both monomial so numbers with numbers letters letters 10 times 2 is 20 see has nothing to multiply by so it's just C so when we're simplifying first step is combine the top to become one bottom become one now we have one fraction with one fraction we can then reduce that fraction we reduce this fraction by looking at the numbers first so we have 5 over 20 if you're not sure what 5 and 20 reduces to you put in your calculator base to go 5/20 math frac it'll tell you that you have a 1 on the top and a 4 on the bottom the way they got that was they divided both the top and the bottom by 5 the next thing you would look at is your C values so we have a C squared on the top a C on the bottom the top has more so the top will end up with a C there was 2 on the top one on the bottom so one canceled out from top and bottom leaving me with 1 on the top and 4 D values there is no D on the bottom so it just be one D on the top your answer would be 1 C D divided by 4 realistically we wouldn't write the 1 so we would probably see D divided by 4 it's unlikely you see it but they might even at write this answer like this 1 over 4 C D so remember when you have a fraction with a number on the bottom and it's just a number you can actually pull it out like a coefficient I don't think you'll see that but you might it's most likely you see this answer C D divided by 4 awesome second question second question is a little more typical of what we would see because it's not just monomials and that first question is all monomials which kind of has its own rules almost any monomials go together easily monomials cancel out easily but if you look at the second question we now have binomials mixed in throughout question all right even though it's a harder looking question we follow the same three steps step one is factor looking at that question there's nothing I can factor the way I know there's nothing that can factor I can only factor binomials so these three red things other things I'm going to try to factor if I look at all three of those things none of the word difference of squares meaning I can't square root them and none of them have something I could divide out of them so for instance like if I look at this one up here the first one I just circled there's no number that could divide out of both terms and there's a letter that could divide out of both terms so there's no greatest common factor that I could remove second step in Peavey's any these are things that cause you to divide by zero remember if you have to explain that on a test that's the wording you're looking for right if my envy view is zero I would say zero caused me to divide by zero that's why it's an MPV that's how you would explain your end all right looking in this question the two places I divide are here there's one division and here there's a second division looking at the first one I have 3 times X minus 1 where I look at 3 first because that's the monomial just 3 is there any way that 3 would change him to be 0 no there's no letter attached to it so 3 is going to be kind of fine on its own the next thing we look at is X minus 1 I asked myself what value would make that zero so if X equaled what that would become 0 if equals 1 that means one of my NPVs is X cannot equal 1 it would cause me to divide by 0 the last one we look here is this 2x it's a monomial and there is a letter in the monomial so I could make that monomial 0 if X equaled 0 you got it so those are the NP B's remember on your diploma exam you cannot miss any and you cannot have any extra so go just like guess or throw them in there like if there was another one right here right here so say that had been something else in this instance we don't check you're thinking of division which we're gonna do in a second we yeah we don't check because this symbol right here is multiplication if that was division that's when we check the top yeah ok we've factored you simplify it sir we factored we had Vivi now we need to simplify when we go to simplify what we like to do is combine it into one fraction first and then try to reduce it so to combine it into one fraction first I want to multiply everything across the top when you're multiplying monomials go together and then you do not expand anything else so when I'm multiplying this together I will do 4x times 5 4x times 5 to become 20x but I do not expand my binomials so I would leave X plus 3 as X plus 3 and I would leave X minus 1 as X minus 1 I only multiply the monomial together my reasoning for doing that is because when I go to reduce remember that you can only reduce entire packages or entire binomials so I want it like that so that I can cancel things out easier on the bottom I'm gonna do the same thing monomials go together so I have a three as a monomial and a 2x as a monomial that would become 6x and then I leave the X minus one I do not expand that stuff out I've successfully written it as one fraction I can now work to reduce that fraction we will start with the monomials so I have 20 and I have 6 both of those numbers could divide by 2 to become 10 over 3 if you didn't know how to do that your calculator does it for you 20 divided by 6 math frac it'll tell you what number goes on the top whatever was going on the next part of my monomials I have an X and an X those X's can cancel because they were both monomial axes on the top my next thing I have is X plus 3 X plus 3 is a binomial you can only cancel with other binomials so other packages there is no X plus 3 on the bottom so that means that X plus 3 will remain on the top it cannot cancel out my final thing that I have is I have an X minus 1 on the top X minus 1 on the bottom those are exactly the same packages so they can cancel out making my final answer 10 times X plus 3 divided by 3 how we feel about that do you remember awesome division so when you're dividing a fraction by a fraction we still follow the same three steps the first step is factor in a we can't factor it and there's nothing that factors out second step is NPV things that cause you to divide by zero I want to look at any spot that there's a division symbol so there's a division symbol here division symbol here and then there's this division symbol in the middle that division symbol means that you're dividing the first fraction by this value this numerator so if that numerator was zero you would then be dividing by zero which is not allowed so it's a third spot you have to check for NPD's looking at that green highlighted spot that I have what value would make that green area equal zero it's you it's a mono meal with a letter monomials with letters that letter can't be easier fantastic that was step 2 in PV step 3 is simplified when you have a fraction divided by a fraction it's really hard so we have a way of making it easier what we do with fractions and fractions as we kiss and flip kiss means you take the division symbol and you change it to a multiplication flip means you take the second fraction and you flip it upside down the first fraction remains the same then you have a multiplication question and now you're just doing the same steps we did in the last questions go across the top go across the bottom try to combine it to become one fraction then reduce that one fraction so if I'm looking at the top I have 5 + ^ 4 times 6 the 5 and the 6 combined to be 30 and then n to the power 4 on the bottom I can't combine those yet because there's a squared right it is a monomial but it's a monomial squared so I actually have to put that squared into the brackets to become 25 n squared and then I can multiply across the bottom so the bottom would become negative 50 and squared I only expanded that fraction or sorry that exponent because it wasn't long on there could have been a binomial I would have slept alone now looking at this numbers with numbers okay so 30 and negative 50 can both divide by 10 we also do something called floating that negative to the top right it's just more standard to write the negative on the top instead of the bottom of the monomial so I would write this as negative 3 over 5 I then look at my letters I have n to the 4 and N squared the top one has more so I know the top one should get the N in the end four versus two two of them would cancel on the top on the bottom leaving me with two left over on the top remember early getting exponent rules that we subtract exponents so if you're still using that method that's fine so you go four minus two is two awesome second example first thing factor already been factored for you second step and PBS NPVs are places that caused me to divide by zero so we have to check this three X this X this X minus two this four and this X plus one and we check all the things that would make those equal zero starting with the 3x what would make 3x equals zero if X equaled zero if I look at my next monomial I have an X again X would not be allowed to be zero because it's another monomial with a letter we don't need to write that twice I don't need to say X can't be 0 and X can't be 0 all right that's redundant so just leave it as writing at once then you're good the next space I check would be X minus 2 it means X cannot equal 2 because this is a division I need to check the second fractions numerator the top there's nothing that will make 4 equals 0 but there is something I'll make X plus 1 equals 0 it would be negative again that would be a wonderful opportunity for the Diploma examination to say what are the NPP's and then it would be explained why negative one would be an Antonov rehersal value and your explanation would be negative one would cause me to divide by zero which is not allowed awesome I don't think you have to go into detail about why you can divide by zero you're just stating that would be okay second step whoa we did take us out fvb third set simplify because it's a division of fractions we kiss and flip kiss means you turn the division into multiplication I'm gonna actually represent multiplication with the dots have an X just because then the X's look the same and you flip the second fraction so it's now x times X minus 2 divided by 4x plus 1 the first one is still 2x plus 1 divided by 3x again we try to write it as one fraction then reduce to write it as one fraction the top one would become 2x X plus 1 X minus 2 the monomial is combined to become 2x the binomials all stayed the same in the denominator the monomials combined to become 12 X and the X plus 1 order main X plus 1 once you've written as one fraction you try to reduce that fraction to reduce that fraction you look at monomials so we have 2 and 12 would become 1 over 6 and the x's would cancel we then look at binomials we have an X plus 1 and X plus 1 those would cancel and then we look at the X minus 2 which has nothing to cancel with so we leave it as X minus 2 again it's pretty unlikely your answer would have that one in the numerator there's a whole auto right up there so instead we would just say X minus 2 over 6 we talked about this last day I think one of the hardest parts of this unit is knowing when you're done because people always want to keep reducing and as the rules get more and more complex as we add subtract and solve it can be hard to know when are you done a question you are done a question when you cannot factor and you cannot cancel anything out so looking at that I can't factor it people would probably be ok with them but can you cancel more out is where people would make a mistake here what they typically would do is they would see this two in this six and the urge is to cancel those out to become one over three because they both divided by two the reason you can't do that is because the two on the top is not a monomial and the six on the bottom is a monomial the two on the top is part of a binomial which is X minus two so what you'll see me doing this unit a lot is I always leave my brackets even in my final answer I leave my brackets so that I don't forget oh yeah that too is part of the binomial binomials can only cancel with other binomials this is my final answer I can't simplify any more especially when it comes to letters that's when people make mistakes had there been another X on the bottom people really really want to cancel those X's out and you're over canceling things simplifying hard because you don't get like a perfect ending point you know when we solve at least we know we have x equals a number and we're done when it comes to simplifying like we have here it's hard to know when you're done so you never get a final answer really awesome that's mall 20 dividing we had done both of those last day so it should have been somewhat review here's two hard ones I'm actually gonna do them with you and I'm also gonna rewrite them so some of these would not be on your diploma exam remember that you will not have to factor a trinomial it comes to simplify that means you would not have to factor this you wouldn't have to factor this this or this those things would all come factored for you so I'm going to change those right now so you could put the new values in your notes and then then we'll do the question together so instead of 2x squared plus 5x plus 4 you would be X plus 4x plus 1 and if you remember how to do that that's fantastic if you don't that's okay I can teach you ways around it where we do use it in the denominator there would be a GCF of 2 so a 2 would come out first and then you would be able to diamond method that so you'd be looking at things I multiply to be 4 and add to be negative 4 so you'd have X minus 2 and X minus 2 fantastic I'm gonna give you a chance to do this in steps and I will do with you the first step to every question is factor there is more factoring in that question so I want you to try to factor it up okay Factory in this looking at the first fraction that wouldn't factor so I would still have X plus 4 X plus 1 and on the bottom I would self to X minus 2 X minus 2 the other fraction though can factor so if I look at the top here what I can do to 4 X minus 8 is I could take a greatest common factor out I could GCF it I could take a 4 out leaving me with an X minus 2 in the denominator I have an x squared minus 1 I can't GCF that but what I can do is I could do a difference of squares it's a difference question because it's subtraction and I can square root x squared and I can square root 1 so this would become X minus 1 X plus 1 again we reviewed how to do that in the last class so if you're still struggling with difference of squares or GCF come talk to me we'll figure it out like you can learn how to factor in an hour of it not even half hour you just gotta come ask for some help awesome step 2 to every question is state your NPV so I'll give you a second to try to do that on your own okay and v's things that caused me to divide by 0 looking at this I want to look at the denominator things that caused me to divide by 0 nothing is gonna make this monomial to equal 0 X minus 2 what Evan NPV of X cannot equal to 2 minus 2 would be 0 0 Islam a lot the other one would also be 2 we don't need to write it again second fraction I have an X minus 1 that means positive 1 would cause me to have a 0 the other 1 X minus 1 that means negative 1 else wouldn't be allowed I typically write it like this positive negative 1 if you put 1 comma negative 1 that's the same answer and you do not need to worry about it I just want you to see that notation because it's maybe been a while since we've done plus minuses step 3 simplify simplify means write as one fraction and then reduce that fraction on the top I can write to this one fraction monomials go with monomials the four it's just gonna be by itself binomials just all get written out so I have X plus four X plus 1 X minus 2 in the denominator monomials go with monomials so 2 is still there they have X minus 2 X minus 2 X minus 1 X plus 1 at that point you can now reduce it because you've written it as one fraction when reducing it monomials go together four divided by two would leave me with a two on the top or one on the bottom you don't need to write that one X plus four x' can only cancel with other x plus four x' there aren't any so i put x plus four on the top x plus one can only cancel with other x plus ones there is an x plus one on the bottom so they cancel out X minus two can only cancel with another X minus two on the bottom there's two X minus 2's if there's one on the top it can only cancel it one on the bottom so it's only gonna cancel I'll pick the middle one doesn't matter what you're doing looking at the bottom we have dealt with the to write the two and the four cancelled out and we have an X minus one and an X minus two leftover both of those need to get written out so X minus 2 and then X minus one and I am now done the reason I'm done is I can't factor and I can't cancel out anything else so those binomials I can't cancel any of the binomials completely out with another binomial I understand that have an X on the top and an X on the bottom but they're all attached to packages right they're stuck with their binomials so unless you can cancel the entire binomial you can't do anything else okay we're gonna try it's equina the second one is a little bit more challenging to factor i will factor the trinomial for you so that trinomial should not have been 12x squared minus 6t x plus 75 I've done it on the board up here really quick if you want to see it should have been 3 times 2x minus 5 times 2x minus 5 that is now completely factored on that table I'll give you a minute to now try to do the first step so factor that entire rational expression question okay if you factor that thing will start with 4x minus 10 for X minus 10 I could take a GCF of a 204 divides by 2 and 10 divides by 2 leaving me with 2x minus 5 in the denominator I can't factor X plus 3 but I am gonna put it in brackets the reason I put it in brackets is to remind myself it's a binomial it's a package I can't cancel it out with anything else then I have divided by it and I've already factored the top for you so it's 3/2 X minus 5/2 X minus 5 again you will not have to factor a trinomial on your diploma exam if you don't want to yeah I'll show you in the future and a spot where we might have to and how you could avoid doing it if you wanted to avoid doing in the denominator 2x squared minus 9 when you go to factor the first thing you always check for is a greatest common factor that thing does have a greatest common factor I could take a 2 out which would leave you with x squared minus 9 if you did that that's wonderful but after you do a greatest common factor you have to check to see if you have among other factor you could do looking at that x squared minus 9 I could do difference of squares on it meaning that I would have the two out front and then square rooting x-squared - that it would be X minus three X plus three those are the tricky ones where you have to GCF it then difference of squares a fantastic step to steep the non ferrous or values this is a little trickier one because it's a revision question too because some of the binomials are hard to tell with the answers I'll give you a minute to try it on your own see if you can state those NPVs all right the non crystal values look here the non crystal values on this guy right here I have an X plus 3 in the denominator that means X cannot equal 3 I'm gonna do it up here I like to always put my NP B's in the same spot so I always know where they are I put them up beside the original question it just keeps my work organized especially have some of these questions that get kind of messy if I put my NPV in the middle the question I might lose it later on so I try to make it nice and clear at the top the next thing I look at is this - nothing will make that to equal zero because it's just a two then I look at this binomial right here I have X minus 3 that means I'm not allowed to have positive 3 as an NP be sure I'm not to have positive 3 as a value for x which makes it then you'll notice this time I wrote them separate negative 3 positive 3 that's fine if you write it as plus minus tree also cool the next binomial is X plus 3 X plus 3 you would not be allowed to have X minus 3 we already accounted for that on the top the reason I'm checking the top is because of this division symbol okay so I need to check this 3 nothing will make that 3 equals 0 and then I have 2 2x minus 5 so they're gonna have the same NPV little ones if you can't immediately see what the NPV is of a binomial what you want to do is take that binomial and on the side save that binomial cannot equal zero so you do that for the ones that doesn't immediately stick out to what the answer is and then you just do like simple algebra like we would for any equality this is just gonna be a dozen equal sign we want to get X by itself so we do opposite operations starting with additions and subtractions so I'm going to add five to both sides this gives me 2 X cannot equal 5 then I want to get rid of the two so I divide by 2 that gives me X cannot equal 5 over 2 if you put 2.5 that's fine as well I would leave my answer as a fraction it's good habit to get into because we typically deal with fractions hmm okay you've now done the first step factor second step and tv-10 move on to the third step which is simplify you need to write it as one fraction we need to kiss and flip this thing so the first fraction doesn't change the second fraction does it'll now be times instead of divided and you know you're gonna flip the fraction so it's 2x minus 3x plus 3 divided by 3 2 X minus 5 2 X minus 5 your next step once you kiss the clip is to write as a single fraction so that you can then reduce it Tori it is a single fraction monomials go with monomials so the twos on the top would become 2 times 2 is 4 then all the binomials get rid note 2x minus 5x minus 3x plus 3 in the denominator monomials go first there is nothing to multiply by that 3 then we have X plus 3 2 X minus 5 2 X minus 5 now that it's a single fraction we can reduce that single fraction monomials go together so we're gonna do 4/3 nothing can divide into those so they stay as 4 & 3 on the top we have a 2x minus 5 that will cancel a one of the other two X minus 5s we have an X plus 3 there is no X sorry we have an X minus 3 there is no X minus 3 on the bottom so the X minus 3 on the top stays we now have X plus 3 X plus 3s just one on the top one on the bottom they will cancel it in the denominator you'll notice we have one thing left we have this 2x minus 5 so we still need to write it and that is my simplified rational expression awesome I don't think we're gonna get through Ashlyn's questions today so we're actually gonna skip this question oh sorry this is the one I don't like anyway this is the one that like when I try to find the pattern I just I have a hard time seeing the pattern it has to do with the colors moving and the shapes changing the answer is marked for you so it is C so if you want to look back on on your own try your best to see if you can figure out what's changing and what's staying the same and how things are changing it's a challenging one in all honesty again it has to do with like the colors how it colors alternate all right we're just gonna skip down here because we need to try to get to your actual expression as much as we can we now we're going to try to add rational expressions and subtract rational expressions which is actually harder than multiplying and dividing this steps aren't gonna change okay so when we go to factor that you can't but we are gonna do something important that important thing in the factoring step is putting brackets that's gonna save us a lot of simple mistakes that might second step NTV's things that cause you to divide by 0 if I'm looking at this things that caused me to divide by 0 would be n cannot equal a negative 4 or 5 awesome simplify simplify means write as one fraction then reduce to write as one fraction in multiplication was easy use right across the top across the bottom to write as one fraction with addition is hard the reason it's hard is you have to get something called a common denominator the way we get common denominators is by multiplying each fraction by what the other fraction has in the denominator that that fraction doesn't have so for instance if I look at the first fraction the first fraction has an N plus 4 I don't think that it has a plus 4 I think it has an N plus 4 right it's missed it has an entire binomial the entire package that means the second fraction needs that entire package so we need to multiply the denominator by that entire n plus fourth you don't just add 4 or x fourth it's the entire binomial and what you do to the bottom of a fraction you always do to the top of the fraction then I would look at the second fraction and I would say this one has an N minus 5 the first one doesn't have an N minus 5 so I need to multiply by n minus 5 to the top and to the bottom if you look you now have a common denominator my first fraction has now become this my second fraction has become this I'm gonna write those a little cleaner so you can see that there have a common denominator the first one in red instead of saying n minus 5 times for n I'm gonna put the mono meal first because that's what we're used to so we have 4 n times n minus 5 in the denominator we have n minus 5 and plus 4 we're still adding the second fraction is now 3 n on the top times n plus 4 divided by n minus 5 and plus 4 you have now successfully gotten the lowest common denominator you didn't have to be Louis just get a common denominator the reason that's useful is I can now write this as one fraction when I go to rate it as one fraction on the top you want to expand okay so I'm looking at the red fraction that means the 4 n is actually going to go into the brackets to become 4 n squared minus 20 m and then on the second fraction I'm also going to expand but because I'm writing this as one fraction now instead of just expanding 3 n I'm expanding a plus 3 n so I make sure I attach that middle symbol to the monomial and then I multiply it in so if I have plus 3 n times n that becomes plus 3 N squared if I go plus 3 n times 4 that becomes plus 12 and when you're combining fractions that are being added together you expand the top but the bottom just combines to be 1 so you would have n minus 5 times n plus 4 you just combine them to become one the reason we do that and I'll do this on the side you don't need to write this if you don't want to if you think about fractions and we're gonna see what basic fractions okay like stuff that we understand so if I have two tenths 2 over 10 and I add 4/10 I would have total 6/10 I would not have 6 twentieths right that doesn't make sense maybe it works better with quarters if I have 1/4 plus 1/4 and if you realize I forgot about that 1/4 plus 1/4 equals 2 quarters right I have 2 quarters I don't have to 8 I don't have two of something else is I have 2 and still quarters that's exactly what we've done with this rational expression when we combined it into one yeah it's a little more ridiculous right we have 4n squared minus 20 n n minus 5n plus fourths sounds crazy if I was to add that to the 3 n times n plus 4 over N minus 5n plus 4 force that would combine to be one thing the bottom doesn't change it just combines to be one awesome we have successfully combined that fraction into one fraction now our job is to reduce it before you can reduce it you need to collect like terms with adding and subtracting there is a lot of like terms on the top so I have a 4n squared and a 3n squared that would add to be 7 n squared I then look over here and I have negative 20 n plus 12n that would equal negative 8 ends and then it's still divided by n minus 5 and plus currently if I tried to cancel anything out I cannot because this top is a binomial that means I could only cancel it out if there was another 7 n squared minus 8 n that I could cancel it with which there isn't however I know I'm not done the question the reason I'm not done the question is because I go to my checklist I ask myself is there anything that can cancel or is there anything can factor I can factor that numerator what could I do to that numerator take an a note I could GCF the numerator and sometimes after use factor as your final step you create something that can cancel out in this case it won't but we should always be doing that just in case so I could take an N out really maybe with 7 n minus 8 and then the denominator I would still have n minus 5 and minus or n plus 4 that's my answer because I cannot factor anymore and I cannot cancel out any of those binomials with binomials adding and subtracting is much harder than x all right let's try a subtraction question first step is factor every single time I can factor that so if I look at it and the first one would become 1 over X minus 6 X plus 6 then I have subtraction 1 divided by and I can factor that second thing as well I'm gonna factor out an X but I'm actually going to strategically factor out a negative X the reason I'm gonna take out a negative X's then I know that it will reverse the order of those things and the reason that would be handy is that would now look more like the first fraction so if I take on a negative X you would reverse the binomial and I'd be left with X minus 6 6 X divided by negative x would be negative 6 negative x squared divided by X would be positive X just on the side in case that's like I see some people looking at that like what the heck is going on so this is what I did I took out a negative X that left me with negative 6 plus X does it change the symbols and then I just rewrite that [Music] GCF again negative does something else handy for me it allows me to take advantage of that floating negative remember I always had negatives on the bottom that are monomials we put to the top there's another place we can put it we can actually put it in front so when I have this negative here you can actually go to the top or the front right there and it can change that subtraction question into a addition question so that I now have this as my problem one over X minus six x plus six plus one over X X minus six if you didn't do that you still a subtraction question that's fine you can do it exactly the extraction question I might be good mr. Bracken all we've done is factored that's it that was the factor step step 2 non permissible values the NPV is on this thing X cannot equal positive or negative 6 there's another NPV what else would it be 0 the reason it's zero is because there's a monomial with a letter monomial was letter right there awesome now we want to combine those into one fraction because that's the first part of simplifying to combine into one part of one fraction you need to have a common denominator we'll start with the X minus 6 I have an X minus 6 in the first fraction I look at the second fraction there is an X minus 6 that's wonderful that means we don't need to do anything with it I look at the first fraction I see that I have an X plus 6 that means the second fraction I need an X plus 6 and what I do to the bottom I do to the top first fraction is now done I look at the second fraction and I notice hey there's this monomial X that means the first fraction needs a monomial X and what I do to the bottom I do to the top I now have covered everything in the denominators to give me X divided by X X minus 6x plus 6 plus 1 times X plus 6 over X X minus 6 X plus 6 I now have my common denominator to combine that to become one fraction I expand the top and then combine it looking at the first fraction there is no expansion needed right this X is all by itself so it's continue to be by itself the second fraction I have this plus one that needs to be multiplied in if you didn't write the one like I wouldn't have written the one normally then all you're doing is putting the plus the positive into the brackets which doesn't do anything but it's a habit we get into because if it was a negative the negative would matter and we would have to put that negative into the brackets so it's a habit I'm just gonna create and continue that I put this plus into the brackets it doesn't do anything it leaves me with plus X plus six but it's a habit I want to create on the bottom I now just write my common denominator as the denominator once you have everything as one fraction you are going to combine like terms on the top there is like terms I have X plus X that would become 2x plus 6 divided by X X minus 6 X plus 6 [Music] I'm done when I can't factor and I can't simplify looking at that top I can factor that I could take a two out making it 2x plus 3 the denominator cannot change nor would we want it to change and that is my simplified rational expression how I feel about that okay remember a lot of you guys took 20-1 fact I think all of you did yeah not all with me you didn't do it so you've done that question like four times now in your life okay I'm gonna rip through this one fast so you can see how quick it goes so and then the next one we'll start the day with it'll be a subtraction question if you're not here which many you guys are gonna be here I would definitely watch tomorrow's video on the subtraction because we haven't really done a subtraction question we did that little trick that turn the negative into a positive but you haven't seen what happens when you can't do that trick so please make sure that you watch that or do that example or take your friends notes or something this one though if you want to rip through these questions first step is factor this factors to be 5x minus 5x plus 5 because that was a difference of squares on the bottom on the top is still 4 that trinomial you would never need to do but it's X plus 5 times X plus 5 you would not need to factor that trinomial on oDesk second step is NPVs X cannot equal positive or negative five third step is simplify to simplify I want to write it as one fraction then reduce to write it as one fraction I need a common denominator this one has an X plus -5 this one didn't so I give it an X minus five this one has an X plus five this one has an X plus five but if I look this one has a second X plus five the first one does not so I need to give it an X plus five what I do the bottom I do to the top this now gives me a common denominator I'm gonna rewrite my work so that it's not so messy I'll put the monomial in the front so if I'm looking at that first fraction I'm gonna put the five out front and then X plus five the denominator I'm gonna write it as X minus five but instead of writing X plus five twice I'm going to go X plus five squared plus 4 times X minus 5 and in the denominator I'm going to go X minus five x plus five squared awesome I got a common denominator the reason I want to come nominator is it allows me to combine them now the way I combine them is by expanding the top so the five is gonna come into the brackets to become 5x plus 25 and the denominator does not change so the denominator will remain X minus 5 X plus 5 squared the plus 5 or the plus 4 will multiply in to become plus 4 X minus 20 and that's kind of screwed up my bottom I don't need to write twice because I was expanding the radius one fraction so I realistically look at it like that now that I've combined interview infraction I collect like terms on the top this would become 9 X plus 5 the denominator stays X minus 5 X plus 5 squared I know I'm done the question because I cannot factor anything and I cannot cancel any binomials out with binomials because remember there's brackets around that top