Stony Brook MAT 123 Fall 2015
Lecture 13: Exponents, ellipses, polynomial division
October 7, 2015

Start   we got wolfstock in two weeks thats a lot of fun for those of you who are freshman the homecoming is a lot of fun we have midterm three weeks from tomorrow you just had a midterm youll have chemistry somewhere in the middle and if your lucky also biology or math, if your really lucky all your finals will be on the same day
0:34that happens alright first lets review our rules of it so this should be review for everyone
1:08okay if had x to the 5 plus x to the 8 what can i do with the powers?
if i add them i would get x to the 13 i get nothing if you add two things like that you cant do anything to it you can factor out the x to the 5, but you cant do anything if youre just adding the 2
1:33if you are multiply them you can add the powers do you know why thats true?
that is true because x to the 5 is x times x times X times x times x the x to the 8 is x times x x 8 times so when you put it together you have 13 x
2:01thats why x to the 5 times x to the 8, you get x to the 13 if you have x to the 5 plus x to the 8, its just x to the 5 plus x to the 8 if i raise it to the power then im getting x to the fifth times x to the fifth times x to the fifth eight times so thats x to the fifth times 8
2:31which is to the fourty so you want to make sure you dont confuse these 3 things the general rule you have x to the a times x to the b thats gonna equal x to the a plus b, it switched up when we logarithms got x to the a
3:01raised to the b thats x to the a b what if i had x to the a over x to the a divided by x to the b so thats gonna be x to the a minus b, why is that true?
suppose we had x to the 8 over x to the 5
3:31so whats going on on top? we have 8x's good thing i didnt pick a big number and on the bottom we have 5x's so i can cancel those 1,2,3,4,5 and left with 3 so thats why you have to do the subtraction you have x to the 8 over x to the 5
4:01so in general we gotta make sure, we remember these rules and you will read those so we have some stuff that has exponents and it could be very tricky but you make sure you know your exponent rule this shows up alot in first semester calc do you a lot of playing around with exponents so far so good?
4:32alright what if i have?
what if i have x to the 5 over x to the 5 well that would just be one so what do i do with those powers well we get 5 minus 5 which equals 0
5:00if you think about it more genrally if i have x to the a over x to thea thats x to the a minus a which is x to the zero which is 1. so therefore any x raised to the 1 is 0, unless x is zero which dont worry about that make sense so far?
you can add that to our list of rules
5:33you have a couple more coming so far so good?
what about if we have 1 over or negative what about a negative power what does x to the minus a equal
6:07correct. nobody heared that but thats okay that would be the same as x to the zero minus a which is like saying x to the 0 over x to the a, given in the other direction if you have x to the 0 over a is like x to the 0 minus a you zwitch x and negative a, since x to the 0 is 1
6:34thats 1 over x to the a si another rule for us what about
7:01okay what about x to the 1/a you know what that one is?
perfect thats the a root technically 1/2, the square root x to the 1/3. cubed root you go all the way back and stop
7:32so far so good?
with those it would be rules of exponent
8:06sure that will go up later, i dont have my glasses on one of the things we want to make sure is the rules of exponents because this is precalculus and here you already know the rules of exponents anyone do the 10th webassign yet?
was webassign due? okay its done good howd we do on it?
badly? well..
8:34well yea ill talk to professor sutherland, its out of 14 it should be out of 12 thats okay we will recover oh good, okay alright
9:00so alright back to what we were talking about the other day remember we talked about circles i didnt do anything on ellipses so an elipse is sort of a stretched out circle its a circle where instead of the then being round its more like that the elipse has
9:33two axis like that so sort of two diameters it has a long diameter and a short diameter so this is called the major axis and this is called the minor axis why is ellipse important, well conterie orbits all ellipses
10:01in case you were wondering people spends thousands of years trying to get conterie orbits to work on circles and along came Kepler and said or you can just do ellipses and that solved the problem made a lot of people feel really dumb thats where keplers law came from, when you take astronomy you learn that and then after that things a lot easier not exactly an ellipse but much much muchh closer so its pretty close to a circle
10:30but the earths orbit is a little bit wider then it is this is wider in this direction than in that direction well that sounds like a lot but whats the circumference of the earths core?
no so its 93 miles from here to the sun, you got to multiply that by 2pi so 150 millione its a little more in one direction then the other
11:00so thats the major x and the minor x, so what the equivalent of the radius is the semi major axis and the semi minor axis call that a and call that b no we originally had x squared plus y squared equals r squared now what happens is you think thats a circle right?
11:31circle if you divide it by r squared you get this little struggle with those twos its goes back to the twos probably not that long so thats another way to think of a circle, notice these are the same radi where an ellipse you get x squared over a squared
12:01plus y squared over b squared and of course we can just the way we shift a circle we could..if i had center h,k a more adavnce version
12:31all im gonna show you about ellipses no need to do more theres a lot to learn about ellipses but its not really our focus of course gonna erase this so for example suppose i gave you
13:07gave something like that we want to just graph it so first we climb the center. the center is the 2
13:34comma negative 1 so you go to 2 and you go down 1 and thats where the center is and its very simple whats the square root of 16 4 you move 4 in the x direction and 4 in each direction and the square root of 9 is 3 so you go to
14:013 in this direction 3 in this direction your ellipse not very hard. so this distance is 4 and this is 4. this because the square root of 6 the distance is 3 3 in the center is a 2,-1
14:34not hard right?
thats all im gonna ask of you of ellipse, im asking you to graph it not asking for completing the graph, whats the equation okay so im just working in the other direction we good with these?
so i want to make sure we got the concept of parabulas you just did the vertex roots and completing the square thing
15:00we did circles ellipses lines now we are gonna do a whole new different kind of function you guys might finish early today, unless you guys want to say to 6:50 some of you complained about that once you want me to tak eit
15:50the negative one, that comes from here so the center is x minus 2 and y plus 1 gives you the center positive 2 and negative 1
16:02because you said x minus 2 equals 0 you set y plus 1 equal to 0 and thats the center and then the square root of 16 equals a and the square root of 9 equal b and remember its x minus h squared over a squared plus y minus k squared over b squared equal 1
16:33and this is an ellipse has a centricity of 4/3 how do we know if the ellipse is tilted the other way?
this isde is greater than this?
can i pull this down?
one hand
17:04what if i did that so its very smilar to the other one except i switched the 9 and 16 now what happens? well i have the center still at negative 2 one
17:30and a is the square root of 9 b is the square root of 16 i go to 2,-1 again this time i go 4 in the y direction that looks about 4 and i go 3 in the x direction and it will look like that
18:02thats when you can tell if its a long skinny ellipse or a short wide ellipse not sure to say fat because we dont use that word how we doing?
what are we confused about?
18:35okay 3 spaces from the center okay we need to move 3 in the x direction and 4 in the y direction from the the center root 2,-1 okay?
none of this will be in the part one of the retake
19:00okay study the exam you get back and look back at the webassigns. the new version of the part 1 could look a lot like the old version if its the same kind of question, gott do it again again. maybe 50 times or once you dont have too, im happy for you everybody ready to erase this?
19:31who needs another minute?
alright something totally different how many of you ever did polynomial division in high school?
20:01or synthetic division? oh yes that. you love that stuff love that stuff okay?
polynomial long division is a tool youll need in calculus
21:07suppose i have no idea why but suppose i take this polynomial and divide it by x plus 1 or another words, i want to know if this polynomial can equal to x plus 1 times some other polynomial because if it is, we know it has a 0 at x equals -1 this polynomial of course would be a quadratic
21:32because a quadratic times the x plus 1 would give you something cubed so lets figure out what the quadratic is, how would i do that i would do that with something called polynomial long division you write this thats the way you wrote long division back in the second grade thirdgrade any grade whatever grade you learned it in. i dont remember when i learned it in
22:04i think around 3rd grade does that sound right?
2nd grade, 3rd grade?
it was a long time ago thank you but it was trying to think when i 2nd grade would of been ten years ago no am i that old?
22:319, 9 that means i graduate high school next year so no matter how you do division oka you take division you divide this into each term you can bring it down multiple and pull it down and factor and do it again so what you do is you only look at the 1st term x and you say to yourself how many times does x go into x cubed
23:00well another way to think about it is x times what gives you x cubed x times x squared gives you x cubed so x goes into x cubed, x squared times and the same way you did the division, you take the x squared and you multiply it by this and you get x cubed plus x squared okay then what do you do with the division, you subtract x cubed minus x cubed is 0
23:315x squared minus x squared is 4x squared and bring down the 7x just the way you do division so what did i do.. i said x has to go into x cubed how many times, it has to go in x squared times x squared times this gives you x cubed so now i took x squared and multiplied it by the whole number i get x cubed plus 1 x cubed from x squared and i subtract
24:02so i subtract x cubed minus x cubed goes away 5x squared goes to 4x squared and you bring down the next term and you do it again x times what gives you 4x squared, 4 x but 4x times x is 4x squared 4x times 1 is 4x and i subtract
24:304x squared minus 4x squared is 0\ 7x minus 4x is 3x bring down the 3 now x goes into 3x how many times? it goes in 3 times so this gives me 3x plus 3 and when i subtract i get 0 which means it goes in perfectly
25:00so you can use this to factor polynomials cause now i can look at this polynmial never done this before? well we will do a couple of these to get the idea im gonna pull this down okay by the way this is accurate
25:34why would this be useful? well go back to the original polynomial, x cubed plus 5x squared plus 7x plus 3 i now know that this has a 0x equals negative 1 a second 0 x equals negative 1 so two of them and a 0x equals negative 3 i can graph it and do all sorts of fun stuff from there or not i vote for not but theres lots of stuff you can do
26:00try another one of these?
26:39rote these down and left them in the office suppose i wanted to know i want to know if this factorable by x plus 2 im gonna to this one and then ill let you guys do one
27:04so i write this as x plus 2 goes in to x cubed -3x squared minus 8x plus 4
27:30so im gonna take x and divide it into x cubed x goes into x cubed x squared times because x squared times x is x cubed now i multiply x squared times x x squared times 2 and i subtract always subtract and that will help you keep track the sign changes so the x squared goes away thats the whole idea and minus 3x squared minus 2x squared, minus 5x squared
28:05bringing down minus 8x and now repeat x times what is minus 5x squared minus 5x so you get minus 5x squared minus 10x and subtract minus 5x squared minus minus 5x squared
28:30is plus, goes away minus 8x plus 10x is 2x bring down the 4 and somebody will look at this and say x plus 2 would work, right?
x goes into 2x 2 times so i get 2x plus 4 and i subtract and i get 0 so this tells me that x cubed minus 3x squared
29:02minus 8x plus 4 equals x plus 2 times x squared minus 5x plus 2 yes?
yea i love when you guys ask me that stuff it makes me feel so inadequate
29:45im not sure what you are saying this one?
can i take x times x squared i get x cubed and x squared times 2 is 2x squared and then i subtract
30:00this is minus 3x ssquared minus 2 gives you minus 5x squared and minus 3 minus 2 is minus 5 lots of opportunities to mess up in a problem like this because subtract now im good back at my inadequates
30:30more questions feel free no love this, should i make one for everyone to do oh yes sure oh yea i have a bunch of thes im gonna make them harder too yea i had them in my office and i left them there
31:32that should work what do we know about practice?
what do we know?
32:01be perfect, be practice go aim hire everytime
32:40that says x to the 4th minus 2x cubed minus 11x squared plus 13x plus 3 now we are gonna divide it by x plus 3 see how we do
33:03x plus 3 now i gave you a slighly hard one for those of you watching the camera go around go the other way so x goes into x to the 4 x to the 3 times
33:35so x to the 3rd times x is x to the 4th x to the 3rd times 3 is 3x to the third and now you subtract negative 2x cubed minus 3 x cubed is minus 5x cubed now bring that down
34:03so x becomes something tha makes 5x cubed so thats minus 5x squared you do minus 5x squared times x is minus 5x cubed minus 5x squared times 3 is minus 15x squared subtract these cancel minus 11x squared minus minus this is a plau
34:314x squared bring that down x goes into 4x squared 4 times and you get 4 x squared plus 12x and cancel i mean subtract 13x minus 12x is x
35:02plus 3, plus 1 then you multiply 1 times x plus 3 and you et x plus 3 so the answer when we said what is this divided by that?
we said what is x to the 4 minus 2x cubed minus 11x squared plus 13x plus 3, divided by x-1
35:34and that was the question and this is the answer plus 3 now, yes?
one more, oh yea ill give you one more
36:02i got two more now what if it doesnt go in evenly?
lets do one that doesnt go in evenly first takes me a few seconds to make one up
36:51alright ready?
what do you mean no?
37:24whoop almost made a mistake
37:30ill just rewrite that because its easier for me okay ill give you 5 mins you ready who needs more time> are we ready?
we ready? we were born ready?
38:03so x goes into x to the 4th, x cubed times x cubed times x is x to the 4th and x cubed times 5 is 5x cubed and you subtract and minus 9 plus 5 4x cubed
38:30bring down the 23 23 x goes into 4x cubed, 4x squared imes 4x times x is 4x cubed 4x squared times minus 5 is minus 20x square woah time out yess yes yes negative 9 plus 5 minus 4 i knew i was in troubke minus 4
39:08when i got the 43 i knew we had a proiblem and subtract and you get 3x squared and by the way, in your brain you say for a second these numbers look weird double check very easy to mess this up x goes into 3x squared 3x times
39:30so bring that down and you get x bring down the minus 5 and thats just going to be plus 1 howd we do one this one? good?
alright now we give you a slightly harder one
40:19one more try x goes into x cubed x squared times x suqared times x is x cubed
40:32x squared times minus 5 is minus 5x squared so far so good minus 8x squared plus 5x squared minus 3x squared bring down that 4x and repeat x goes into minus 3x squared 3x times
41:04you subtract and you get minus 11x minus 1 is that correct,we all happy with that so now x minus the 11x is minus 11 times and then you multiply and get minus x plus 55 or i would like to say fifty five
41:30subtract you get minus 56 but wait a second its suppose to be 0 so what do you do here to get minus 56, thats the remainder so the remainder, just the way it is in division so if you took x minus 5 and multiplied it by this you will get that so thats your left over so it doesnt always go in smoothly
42:02so sometimes you get a remainder that remainder is very useful if you want to find the limit you can write minus 56 i got one more for use to solve how would the answer look? this would be the answer
42:33so its this goes into this and you get that okay so if this were a webassign thats what it would look like, when you get this on webassign youll get a couple be careful its very tricky to answer them you have to get it exactly right now we have one other type of thing we can do what if we are not dividing by something linear
43:03you guys should be pros at this by now you get up to now you use synthetic division but now
43:30how about that?
same technique expect whats going in now isnt linear its squared so it gets a little messier then when i did x squared plus 5 x squared goes in to x to the 5 x cubed times x cubed times x squared is x to the 5 x cubed times 4 is 4x cubed
44:08now when subtract you get x to the 4 3x cubed minus 4 is minus x cubed
45:59now x squared times what is x to the 4th? x squared times x squared
46:05so far so good?
so x squared times x squared is x to the fourth x squared times 4, plus 4x squared no when you subtract you minus x cubed comes down and 7x squared, we should of brought that down 7x squared minus 4x square
46:333x squared so maybe you shpuld bring down the minus 4x x squared times negative x will be negative x cubed and youll get negative x cubed minus x times 4 is minus 4x when you subtract you get 3x squared
47:02plus 12 now we just multiply by 3 anybody successfully get that some of you, yes very excited