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TuringTestBest ResponseYou've already chosen the best response.6
FOIL Inopeki :D (a+b)(c+d)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
Good, (no need to write the 1 though) (3p+t)(2p4t)
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Oh right 6p^2+3p*4t+t*2p+5t?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
not 5t... everything else is right though
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
You don't see it? (4t)(t)=4t^2 look at the last terms in the parentheses
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
so you have what then, before simplifying?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
can you simplify it?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
6p^2+3p*4t+t*2p4t^2 6p^2+6p^24t^24t^2?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
no... how did you the p^2 in the middle? we had 6p^2+3p(4t)+t(2p)4t^2 parentheses look cleaner btw, you should use them here what is 3p(4t) and t(2p) ???
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
right, so then we have...?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
yes :D and how does that simplify?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
I dont know actually, maybe 6p^214pt+pt4t^2?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
they are like terms in the middle 12pt+2pt=10pt this goes for any combination of variables a^2bc+3a^2bc=4a^2bc wr^4dt^3+3wr^2dt+6wr^4dt^3=7wr^4dt^3+3wr^2dt notice in this last example we couldn't combine all the terms because one had different exponents. We can only combine when they are the exact same in exponents and number of variables.
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
But dont i need to multiply them to get 10pt?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
no, just add the coefficients factoring is one way to see it... factor out 'pt' from the middle two terms: \[6p^212pt+2pt4t^2=6p^2+(122)pt4t^2=6p^210pt4t^2\]
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Oh wait, i get it now, it would me multiplication if it was 10pt^2
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
not necessarily. You can't multiply things unless the equation tells you too, or you do a trick like multiply both sides if you have 10pt^2+7pt you cannot simplify that...
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Oh ok. But what i meant was that if it were to tell us to multiply it would look like this 10pt^2 and not this 10pt
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
if you multiplied you get \[(2pt)(12pt)=24p^2t^2\]
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Since you would be multiplying both p and t with itself
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Whatever, were getting off track :)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
not really, you need to get this down too. It's all related, and it's good to know where you're at. (2s^2+p)(s3p^3)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
hey I'm green! never been a mod for math before :)
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Congratulations! You deserve it :)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
thanks :) I can't delete things though, don't see the point oh well... (2s^2+p)(s3p^3)
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
(2s^2+p)(s3p^3) 2s^3+2s^3(3p^3)+p(s)+3p^4
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
so far so good now try to simplify...
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
oh wait, small error
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
2s^3+2s^3(3p^3)+p(s)+3p^4 ^^ should be???
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
2s^3+2s^2(3p^3)+p(s)3p^4 ^^ plus that :)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
good catch I missed the sign lol !!! yes excellent now can you simplify it
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Thanks :D 2s^3+2s^2(3p^3)+p(s)3p^4 2s^36s^2p^3+sp3p^4?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
perfect last one: (a+b)(ab)
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
a^2ab+abb^2? a^22abb^2?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
not quite, I don't think those terms in the middle add...
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
The simplification or the first one?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
a^2ab+abb^2 ^^^^^ what is x+x ?
 2 years ago

pratu043Best ResponseYou've already chosen the best response.0
Some kind of coaching class going on here?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
And that is a very important formula @pratu Inopkie wants to learn Algebra, a good cause in my book :)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
Thanks but I think I have an idea what I want to show him, we've been working together for a while ;)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
@Inopeki We just showed that\[(a+b)(ab)=a^2b^2\]That relation is called the 'Difference of Squares' formula. It is used all the time, as you will see, as a quick way to factor anything of the form\[a^2b^2\]
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
Yes, it's always true for that form. This is handy as you will see because if you remember how factoring helped us solve other equations, it will do the same for us here: For instance\[x^24=0\]we can factor this..\[(x2)(x+2)=0\]which as I mentioned befor by the 'zero factor property' means that we need to solve\[x2=0\]and\[x+2=0\]to find the zeros of our polynomial
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
But isnt (x2)(x+2) 0 because the 2s negate?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
foil it out and tell me yourself
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
x^2+2x2x+2*2 Well pellet
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
right which simplified too what?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
so that's not always zero, is it? if x=1 then f(x)=3, not zero what the expression f(x) is equal to here depends on what we put for x. There are in fact only two values for x that will make f(x)=0 Which should make sense by what I said yesterday: The order of the equation is 2, so the total multiplicity of its roots is 2 (fundamental theorem of algebra)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
yes, sorry, bad notation on my part f(1)=3 f(0)4 etc.
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
But how does that work?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Functions, maybe im not ready for that subject
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
No you are, perhaps I haven't been clear enough whenever I write x^24 If I want to know where this graph hits the xaxis it is useful to call this a function f(x) here we have f(x)=x^24 Think about a graph: When the graph is toughing the xaxis it is at y=0, correct?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
I just 'called' it a function, and made it graphable...
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Yeah, when x and y intersect its 0,0
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
f(x)=y+2x3 So if we want to figure out y we need to plug in something for x
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
well, wait now, if you have a y in the function then it is not f(x) here's one thing that I bet is confusing you: f(x)=y when we write a line equation y=mx+b we are saying something about the graph: that the y coordinate is proportional to the x coordinate in this way. However this is rather simplistic. It is really more that we have a function f(x)=mx+b and we have chosen to graph it by letting the ycoordinate of our graph be f(x) they are effectively the same here, but a function can be graphed in many different types of coordinates, so y would not be used unless we are in Cartesian coordinates as we are now.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
hence it's better to think of graphs in terms of functions, represented in Cartesian coordinates by y
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
What are cartesian coordinates?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
the regular ones you know xaxis and yaxis
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
there are other coordinate systems like polar, spherical, and cylindrical coordinates If we tried to graph our function in polar coordinates, y would have no meaning. however the zeros of the function are independent of how we choose to graph it. The math will be the same. Hence when we want to know the zeros of some polynomial we call it f(x), and that allows us to put numbers into it and see what happens f(x)=x^24 f(0)=04=4 f(1)=14=3 f(2)=44=0 so we found a zero simply by calling this a function and putting numbers into it.
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
One question, what do we gain from functions, what are the numbers on the RHS?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
which numbers on the RHS?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Like f(0)=04=4< ^ 
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
well that's the simplified form of f(0): f(0)=4 check it by plugging in x=0 into the function
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
But if we know that f(0)=4 why do we do it?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
lol, why /don't/ we want to do that?! Almost all things in physics can be represented by function. If you want to know when the water pressure of a system will become too high, what trajectory to shoot your spaceship, or where a quantum particle will be at a certain time, you must analyse functions in this way.
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Then teach me! This is important!
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
Really, though of course actual physics problems are usually (though not always) a bit trickier than just plugging in a number but yes, functions are instrumental to calculus, and therefor physics As I said earlier calculus is about analyzing functions... I guess we should start to talk about functions in a very detailed way.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
the definition of a function for you will be: if you have something like y=f(x) (where f(x) is some arbitrary polynomial with x's in it) and for every x you put in you get exactly one number for y back, then y=f(x) is a function, and y is a function of x try to soak that in, I'll tell you more about it momentarily
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
So what you are saying is that if f(x)=y x=y?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
What you are saying is that if x is one and y is 2 and you add 1 to x you add one to y aswell?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
no, just that we get only one value of y for each x for instance f(x)=y I defined that, whether or not it's a function however if for every x we put in we get only one y back, then f(x) is a function, and the variable y is a function of the variable x
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
scratch the *for instance :/
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Gotta go eat dinner, brb
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
look at a graph like a circle (I know you haven't studied them much yet this way)dw:1326043068666:dwthis is not a function because there is more than one y for each value of x... see you in a bit, I'm gonna have breakfast.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
drawing a line from x to its corresponding y value you should know we need a vertical linedw:1326043155512:dwsee how one xcoordinate corresponds to two different y values y1 and y2? But what about y=2x+5 ?dw:1326043286670:dwthis graph has exactly one value of y for each value of x, so it is a function.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
the idea above is what's called the 'vertical line test': if you can draw a vertical line at any point on the graph and intersect it twice, then it is not a function.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
when you return look at this say we have a table of values for some y=f(x) and we want to know if f(x) is a function.... look at the table x y=f(x)  1  2 3  7 2  0 5  2 Is this a function? why or why nmot?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Its not. EXEPT the one that says 2  0<< cause that means that there is only one value of x ^
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
The others seem to have two values of x
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
actually look at the x's each x has one y, so it is a function we have that f(5)=f(1)=5 so two different x's give the same y, but that is okay the only thing against the rules would be to have something like this x  y  1  3 2  5 1  4 because this means that f(1) has two values: f(1)=3 and 4 so this is not a function... look at the x to see which gives more than one f(x)... (I am using the terms f(x) and y here interchangeably) x  y  2  0 3  1 5  6 3  6 1  0 function or not ant why?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
That's right FFM does the curriculum meet your approval? I doubt it with your standards :P
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
x  y  2  0 =f(2)=2? im sorry, i dont really understand 3  1 5  6 3  6 1  0
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
f(2)=0 that means when x=2, y=0, yes?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
that's all there is to that so is there any f(x) on the list that gives more than one number for a single x?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
So, x  y  2  0 = f(2)=0 3  1 = f(3)=1 5  6 = f(5)=6 3  6 = f(3)=6 1  0 = f(1)=0 So, no.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
actually what is f(3) ?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
exactly, so not a function :)
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
f(3)=1+6 so its not a function!
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
And that means that f(3) could be graphed in a circle?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
right not 1+6=7 of course, it equal the set\[f(3)=\left\{ 1,6 \right\}\] It doesn't mean it would be a circle but it would fail the vertical line test, watch...
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
thanks FFM, praise form Caeser :) dw:1326046327322:dwnow look at the points of our possible function above. is there any place we can draw a vertical line that would intersect two points?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
lol *Praise from Ceaser
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
no way to draw a vertical line and hit more than one point you say?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
more than 3? You only need to hit 2 points to show that this is NOT a function and here it is:dw:1326046665544:dwso again we show by this that f(3) has more than one value...
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
are the following functions?dw:1326046773020:dw(I'll fill in different graph above)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
yes f(x)=y because we have defined it that way now:dw:1326046848239:dwfunction?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
draw the vertical line that hits more than one point on the graph if that is true
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
No wait, there are no points
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
there are infinite points on a line.... a line intersects another line at a pointdw:1326047055394:dwP is our intersect point can you draw a vertical line that hits the curve at more than one point?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Oh, so y is a vertical line?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Then this could be a function
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
dw:1326047157974:dw not function
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
now you got the idea :Ddw:1326047282819:dwfunction?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
just one line is sufficient, but yes :)dw:1326047413725:dwfunction?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
good :) x  y  1  3 2  3 3  3 4  3 function?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
But if it was 13 14 It would not
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
good x  y  0  3 0  2 0  1 0  0 I think you already proved you know this, but function?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
nice job, exactly slightly different: f(x)=x+3 function?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
I guess that depends on what you plug in for x?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
It doesnt show multiple values of y, just x+3
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
perfect observation :) every x we put in gives us only one y. y=0 is y a function of x?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Only if we know that 0=x
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
not so, plug in various x into the function f(x)=0, we get the table x  y  0  0 1  0 2  0 3  0 and so on... and I think you would agree that this chart constitutes a function.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
tricky one, but now you know ;)
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Yes, that chart constitutes a function. But if there were no chart?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
They are all in agreement, the chart, the graph, everything. we don't need x=0 to get one value of y back; we always get one value of y. The same value of y, namely zero. do you know what the graph of y=0 looks like?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
yes but I don't want the point (0,0) y=0 is a line: it's actually the xaxis the line x=0 is the yaxis think about that for a minute
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Oh, right. dw:1326048543616:dw
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Didnt really think of that
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
exactly, and does the graph of y=0 pass the vertical line test?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
are you saying that you can draw a vertical line that intersects it twice? show...
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
OH you mean that? Never mind lol
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
so is the graph indicative of a function or not?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
perfect, what about the line x=0? (standard cartesian notation here , don't get hung up on the names of variables, we could easily have them to t and p or whatever)
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
yes, so is it a function?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
be careful can we draw a vertical line that touches the graph at more than one point? what would the table of values look like?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
No wait, this becomes f(x)=x right?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
no, this would be f(x)x=0 (read f of x such that x is zero) here f(x) takes on a range of values from negative to positive infinity: x  f(x)  0  0 0  1 0  2 0  1 0  2 etc. not a function, right?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
I think i get it, its because there is no y?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
but f(x) has endless possibilities
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
no this chart is the same as x  y  0  0 0  1 0  2 0  1 0  2 I told you that y and f(x) are interchangeable y is the name of the coordinate which we are using to represent f(x) so here they are the same the chart itself is of the same type as you saw before, we have one value of x corresponding to multiple (in this case infinite) values of y, so it is not a function.
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Ohhh, but if you had y corresponding to multiple values of x it would be a function.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
let me try to clear a few things up... what is the dependent and what is the independent variable here?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
The dependent variable is y because if its a function or not depends on y having multiple values corresponding to one value of x.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
right, but I'm not sure about your reasoning, so what about f(y)=x=y^2+3y which is dependent and which is independent?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
x is the dependent one i think
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
exactly, so here x is a function of y we could have therefor more than one y value for an x value in this case, but not more than one x value for each y, if it is to be a function. so really the rule is better stated this way: "a relation is a function if each value of the independent variable corresponds to exactly one value of the dependent variable" because we can change the names or swap positions of the variables
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Because we can have more of y but not more of x
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
exactly, look at the new version of our rule above (in quotes)
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
Yeah, that sounds right.
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.6
so whenever we see f(t)=r for instance which is dependent and which ins independent?
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
We should move to another "thread" this one is lagging for me
 2 years ago

InopekiBest ResponseYou've already chosen the best response.0
r is the dependent and t is independent
 2 years ago
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