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2.3 Remainder & Factor Theorems & Rational Root Test

Added Sep 20, 2015 •

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This is a video of me teaching the second half of section 2.3. Watch the ENTIRE video and then complete the 7 homework problems at the end of the video.

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hi students please take out your smile notebooks and add these notes to today's date today's topic is 2.3 will be talking about the remainder and Factor theorems and the rational root test three objectives for today the first objective is I will use the remainder theorem to evaluate a function any given value second  I will use the factor theorem to deter if x minus K is a factor  by evaluating  DePaul at x equals came and our third objective is I will use the rational root test or rational zero test to determine the possible Zero's of a polynomial we're going to learn about that part later but it's ax equals plus or minus.  Set once you have that written down you can go ahead and continue on you might need to press pause do you need two pots  the first we have the remainder theorem and look at number 36 my book it gives you the function G of x equals 2x + 6 + 3x - 4th minus x squared plus 3 so what happens is evaluate gf2 I went too fast and go back  set a value ate two we're going to use synthetic division and what we do since it's two we looking at we put our two outside here and then offer coefficients go on the inside and play Drop It Like It's Hot goes down and then we multiply Singapore and then we narcoms multiply 4 * 2 is 8 and then you add your accounts 22 24 Play 3 in an ad and then song and we 175  that is our remainder right there so are remainders 175 which is what now just for fun we could go ahead and plug in to back up for RX value and we didn't up with two to the 6 or 6 for so we have 2 * 64 + 3 * 2 to the 4th would be 16 - 2 split is 4 + 3 so if we went ahead and we added a laugh seaweed end up with a Cheatham 64's 128 + 48 Swan which is what do you know 175 so it works out so you can use synthetic division to evaluate a function like you to try one now so I'm going to go ahead and give you one to try go ahead right now see if you can evaluate what G of one is all by yourself pause in this video and try it okay let's see how you did should have started off synthetic division same numbers we had a bath  how positive one of the outside because that's what we're evaluating G of 1 stop it Like It's Hot and then multiply and he multiply multiply add and we get 7 so you can get 7 Mansion hopefully you did  now the other thing you can do is you can evolve it on your calculator too if I went ahead and I punched in the function into my y 1 graph and then I did II calculate find the value and I did the value a Nexus One who next is what it is why is 7 so you can check it by plugging it back into the original you can do synthetic division snow way you can miss this problem  let's go ahead and look at our next section is my picture of a squiggly line to separate this section from the next now we're going to be looking at the factor theorem so a polynomial FX Has a factor x - K is an F of k equal 0  so let's see what we mean by that's where I look at it Apple so this is the directions from our buckets as an exercise is 39/40 to use synthetic division to show that X is a solution of the third degree polynomial equation series out to factor the polynomial completely so I'm angle is we need to factor the polynomial completely list all this real where to find the zeros so number 40 gives me X cubed minus 28x minus 48 equals 0 and value of exit tells me is x equals negative 4  says to use synthetic division show that does x equals negative or is a solution so they're telling me that that's one of my zeros so if that's a zero then that should divide evenly and to my polynomial function so let's do some division to see if that really does divide a negative  I thought your coffee chance inside so who won there's no x-ray terms are we ever 0 negative 2848 It Like It's Hot and look what do you know he's got a zoo there which means that that is a fact -4 is a factor of that polynomial when an evenly there's no remainder so we can go ahead and write at what we have left because this was an X Cube play drop it down you lose one so you have x squared leftover - 4 x - 12  he goes there so if I go ahead and list the there is I can factor out that new a quadratic that I have when we can Factor quadratics just to trinomial x - 6 x + 2 should do it and I plus our original factor that we would have loved to give us a fact that gave us the zero but if we know the zero we can say the factor is next it's going to be plus to be nice if the zero is -4 than the factor has to be a plus for the opposite signs so those are my factors if I X 3 of those things out the long way I would end up getting that same polynomial function right there so we did most of what it's a sweetie have factor the polynomial completely but now we have to list out all the real zeros of the function well that's not going hard now that we have the factors zeros you would just set you to factor equal to zero so we're going to have exical six- to in the four let's go ahead and look at our next example will do one more like that see if you can your one out on your own here so this one gives us up new polynomial function and our values of X are value of x is 1/2 so go ahead and prove to me that one half is in fact a zero of the polynomial to do some that division to show that that goes in there so with pastas video and see if you can do some that a division to show that 1/2 really is a zero of that okay hopefully you paused it and I ready to now check your work so let's see we've got the if we prove that there was a zero hour that one half does equal 0 when you do synthetic division so you put your coefficient inside Drop It Like It's Hot multiply so then you have the two times one half was the one at your call him so you can make team 7 2010 and hopefully you didn't see where you went wrong so now we have to use this to factor that this polynomial will this was an excuse and I dropped this down you came to x to the second power so let's write that attitude - 14x + 20 equals 0 so now I can go ahead and Factor two out of that so that's when - 7x + 10 it looks like that China factors mourned x - 5 + x - 2 so now I'm supposed to list out all of my factors so I know I have one of my factors is right there what's -5 another one x -2 night should be got an x to the third power so I need it right now I've just got an x x and x it should be next glitch I know I need to know what least one more X here turn this was a zero at 1/2 so this has to turn into a factor so if 1/2 was a zero than my factors going to have PC I'll be dividing by that to the twos and front 2X and then since this is a positive and here goes a negative Chris voice opposite setting that equal to 0 if you set that equal to zero you would add the one over and then divided by 2 so you would get one half tractors and then your zeros once you have your factors that are easy to say everything equal to zero and solve x would be five positive 5 is it a 2 and 1/2  right let's check out our next section we're going to have  say says in exercise 43 - 48 verify the given factors of the function find the remaining factors use your results to write the complete factorization and list all the results sounds like a lot so let's just start one step at a time self let's look at number 4444 gives you a polynomial function 3 x cubed plus 2x squared minus 19 x + 6 + see a factor of x + 3 so now the other last time it was giving you the zero here this time is giving it to you and Factor so the first part part A says verify the given factors so we have to verify the x + 3 is a factor have our polynomial here which means x + 3 needs to go in divide evenly into this polynomial this is a little bit trickier we still do sometimes synthetic division but because it's giving us the factor instead of the zero over here if the think are we going to put a positive or negative through this time if it's a zero just put it what it what it is but when it's in parentheses you going to do the office so there's a positive to hear you have to put a negative three are the coefficients of your polynomial still stay the same as a 3 8 - -19 + 6 okay so we were at we had executed we dropped it and we had 3 x square now so next Square we should able to factor that further but we did what part it asked parties and verify that it was the factor because they were there so yes three is a factor  a Part B Part B says to find the remaining factors  okay what did you that we have to take what we just found  Destry -7 and 2 and write it as a quadratic equation so we've got 3x squared minus 7x + 2 equals 0 so we can go after that  Quadra equations we can typically Factor  and we can in fact that we could always try to quit at formula to solve for the zeros don't have to do that in this section yet we'll get there though so we've got to factor that without a 3X and an axe  c - 1 -2 give me a -6 and I might as well get that gives me and my 7 in the morning that's good  so then my remaining 6 - 1 x - 2 - Part B is complete C part Cena says to use my results to write the complete factorization so now I have to factor that polynomial completely  so I take f of x + 3x - 1 x - 2 what is the two factors that I just had from right above but we need to listen to the other factors which we already know the fact the other factor that we had was x + 3 so we just scary that factory down and now it's completely factored and if you multiply This Out 3 x x x x x you would see that you would have a 3X Cube as your first term and you could go ahead and look but all at the really long way and you would end up with that whole polynomial  in part D is stop the zero swell once you have the factors the zeros are easy all your factors equal to zero and just Celebrex so we're going to get x equals a positive one third from this Factor this next one's going to give me a positive to  and then this last one should give me a negative three I didn't write a negative Theron you just threw in the negative if we sketched the graph here used as most to grab that out you can see their negative three 323 which surrounded here but three seasons is his areas with the same as 1/3 and 2 so you can say we really do have those three also just a little sign out notice that over here we have an odd an odd degree pawsitive so look at the end behavior the side is going down and the side is going up  okay try on problem  same type I want you to try one more of those freaky one so we've got set timer number 46 f of x equals x to the fourth minus 14 x cubed minus 71 add - 10x + 24 + the factors not gives you two factors or x + 2 + x - 4  and we're supposed is it at same process we did last time we need to verify the factors so we have to show that they exposed to an ex for both factors so we set our division of takeout coefficients put them on the inside but now to show that costumes of factor out here we're going to put a positive or negative hopefully you thinking opposite signs what's on the outside then we drop the heat Like It's Hot multiply multiplies to that what are you makes messed up sorry I got x + 2 is  spell remainder of zero. Just prove that x + 2 as a factor so that x -4 is a factor so we need to put a positive for out there cuz I can't go to the office and signed and we're going to do some thetic division again on the same with the same numbers we just used here so we drop the 8th Like It's Hot 8-12-4 went in evenly to that so we know the x + 4 is also Factor  Part B we're looking for their remaining factors so I just was left with this but keep in mind we started with an 8 x to the fourth so when I dropped you I had an 8 x to the Alyssa be an 8 x squared now so we need to read that 8 quid plus 2X minus 3 equals 0 should be able to factor that what is a quadratic typically these factor in this type of problems  so Factory and there's our remaining factors parts to factor completely so we have to write the polynomial outcome so we take the two factors we just found  and then we have to take our original factors that X+ to from up above and then also are x -4  don't you just think about that for a moment if I multiplied all of the leading term C of 4X x 2 next x x x x that would give me an 8 x to the fourth power and if you look up here we would have an 8 x to the fourth power okay so next  once he finds a complete factorization now you have to find the zeros that's easy Once you have two factors you get x equals negative 3 plus positive 1/2 and positive for all you do is you look at each of your factors and set them equal to 0 make and we can grab it here's a little picture what the graph looks and if you check  if you check out the different zeros you can see the graph touch that -2 + -.7 3.5 or 1/2 and then over here at 4 so your graphing calculator will confirm that you did the problem correctly okay moving on to our last section  the Rational Zero test  sometimes called the rational root test so the Rational Zero test gives us possible rational zeros play to consider an option it'll be in the format says x equals plus or minus p over Q  where P stands for the factors of constant term in the constant term you can think of as the last turn  / the factors of the leading coefficient which you can think of as the first so I was think of it the factors of the last over the factors of the first and then I'll make more sense once we see it in action here so the next section in your book is going to can you use the Rational Zero test to list all possible rational zeros of f and then find the rational zeros  so number 54 exam give this a function f of x equals X cubed minus 4x squared minus 4x + 16 and we're supposed to find the possible rational zeros so what we have to do if we have to list out the possible factors so the first I care about is our last time we went to find the possible factors of my last time are my constant term for 16 is the constant term in this polynomial so what are the factors of 16 Singapore all the numbers that go divided into 16 evenly so one goes into 16 - 4 8 + 16  and then we go and we look at our first term are leading coefficient so there's no invisible one there please put a one there so the factors of 1 are just one down below we just have a 1  semi possible rational zeros is polynomial r + or - 1 + 2 - 2 + - 4 + 1 - 8 + 1 - 16  you just divided each of the terms by whatever's in the denominator  so I know that I have see I guess I'd be at 24682 possible zeros to consider and I also know that there's no more than three zeros because x to the third will tell me that I can't have one in three zeros whatever wake Ria's are not having more than that many  so I have to gas of these 12 which is at least one that divides evenly into into my polynomial  so if I want to find an actual zeros  we start guessing  now want to save yourself sometimes we can use your calculator to help with Mike good guesses  so I grafted look at my calculator to help me make a good guest plug in my  constant or my  coefficients from a polynomial here one- or negative 116 shut the lights number because I got a remainder of zero so I know that too is a zero of that polynomial since it divided in evenly and then that leaves me with an x to the what is wax cubes and I'm left with an x squared Asterix -8 that factors nicely and 2x - 4 + what's team so now I know my zeros are from this first Factor here and have a positive for play all the two the two came from right here is cared that down and then now I'll have this for and then negative too long alright we're going to try one more  okay so find real zeros of the polynomial  looks like we have two more going to try we got back with X to the fourth minus X cubed minus 29 squared sex - 30 cases but in order to begin somewhere so the possible rational zeros are plus or minus P of akhil  PSR factors of our last chance so we have to find the factors of 30 let's see if one goes into 32-3 app for Note 5 app 6708 nope can you tell me at 11 12 13 14 15 yet and then 30  and then the factors of my first time which again there's just invisible One National put it one after the one I just won  Subway Sir singing of all this different choices and in order to save myself some time I'm going to graphic so a graph on Desmos so if we look here I've gotten zero looks like maybe in this was to choose to 4  be at 5 in the hair looks like at 6  so let's try I'll try 6 as my first one and six will be a good one to try to because it's on my list as one of my possible there is there if it's not on if I thought it cost there add like four and a half ice that would waste my time I shouldn't even try van half cuz one half is not on my list  so what are coefficients Don't Talk Like It's Hot  and again look what we have we have remainder of zero so is 6-0 of my pain oh yes it is cuz it went in divided but now it's can I factor this part here if I wrote it all out let's see this was x to the fourth appear so I dropped it on next Cube I don't want to stop at eschewed because I can't Factor those easily typically I need to get down to x squared so I need to find another that divides Emily and I can stay over here it looks to me like I had one at -5 so I'm going to try that try -5 Drop It Like It's Hot and I can -5 was infected that was on my list and it worked out so now I'm down to NX which is good  because one I could solve for that have fun if I saw for that I would subtract the one over which being negative form and then I'd be square root in an egg number what you give me an imaginary solution and this particular these directions here asked me to find all the real zeros so I don't need to list out in majun any zeros here so  turn no real zeros for that so all my real zeros are the two that I picked to begin with which were the x equals 6 + -5 I've too busy Rose and actually this right here waiting up giving me to imagine his ears but we're going to get to that part another day what's happening this time  same directions leave x equals 4 to the fourth minus 43 x squared minus 9x Plus 90 and we're supposed to find all the zeros but where do you start what was the possible zeros so it's your factors of your last time over your factors are your first check so factors of 90 there's a lot of factors of 91 two 3s for Note 5 yes  6 gifts 70899 yes 10 years 15  3045 and 90 and then we have our first term here for to the factors of horse this the first time we've had in a number other than one here so factors of four one goes into myself one two and four so if I were to list out all of these this would take me a really long time a question minus one cuz I have to go one over one and then I'd have one or two and then 105 or  and then I take my next time I take the two and go to over one then two or two but that's 2/4 same as 1/2  so I don't need to say - 2/2 is 1/2 or Force 1/2 because I already have wine 1/2 in the list and then I will go to the next one three everyone is 33423 and so on okay but you think you get if you don't need to list out every single one of them here I just need to figure out of all those choices which one does it look like on my grass and I do I cry so I'm going graph it on Desmos here so there's what does most looks like and sees when I'm counting by here so let's see Alex to me  no that's my access there so right here it looks to me like this is my x axis it's hard to see that but it looks like it touches that naked of to sew and -2 is on a list of possible rational zero so I'll try that so negative to plugged in all my coefficients again the four Drop It Like It's Hot an against looks like it cost on Minecraft and it does cuz my remainder is zero this was a 4 x to the fourth power so now I have to this become a for xcube so I need you to find one that wear crosses and it looks to me see maybe that was Italian by one's there so it looks as if I was at 1 tree looks to me like it's touching at 3  so let's try 3 get 3 is one of my possible zeros makes sense for me to try that one  if I thought that maybe it cost at 4 I would know that that can't be because for not possible zero on my list  I so Drop It Like It's Hot some to fart again can you put  so then it goes in evenly so you get a remainder of zero and then we can go ahead and Factor this last time so we're going to hang to factor that 2X to see what gives me 15 when I multiply but for when I  see you at work to x + 5 -6 and a positive 10 is a positive for so that works out and again the size asked me to find the zeros not the factors but I need to find Susan order to find address so my zeros will be a negative five hats Three Hats and then don't forget your zeros that we picked originally so we have a negative two and a three that's how you find the zeros of a polynomial function  so now I have 7 homework problems and I want you to try so get out of peace paper for your homework  here are you selling colognes stats on page 128 and you have a sample for every single one that I assigned you so use your notes to help you don't do the seven problems and come to class with them done tomorrow might work hard

Uploaded by Kristin DeWit.

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2.3 Remainder & Factor Theorems & Rational Root Test

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