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- anonymous

TuringTest, want to continue?

- jamiebookeater

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- TuringTest

lol I like the attitude, I better eat though, give me a bit
what time is it where you are?

- anonymous

9:45 pm, you?
Then go eat :)

- TuringTest

alright, just so you know it's easy to send messages via fan message now, so use that instead of a question to hail me

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## More answers

- anonymous

Oh, ill use that!

- TuringTest

see you in a bit

- anonymous

Alright

- anonymous

How do you use those fan messages Turing?

- anonymous

Go to the profile and click "write a fan message"

- TuringTest

factor
21x^2+14x

- anonymous

GCF of 21 and 14=7
GCF of x and x^2 is x
21x^2/7x+14x/7x=x(3x+2)?

- TuringTest

almost but
1)be careful of notation
21x^2/7x+14x/7x is not equal x(3x+2)
2) remember to keep the whole GCF outside the parentheses

- anonymous

But i thought that if the 7 was on the LHS then it couldnt be on the RHS?

- anonymous

21x^2/7x+14x/7x=7x(3x+2)?

- TuringTest

but they are not equal
21x^2/7x+14x/7x=3x+2
not 7x(3x+2)
so you must write
21x^2+14x=7x(21x^2/7x+14x/7x)=?

- anonymous

I dont know...

- anonymous

7(3x^2+2)?

- TuringTest

you almost have it but you need to put the GCF outside the parentheses
what is the GCF?

- anonymous

7x

- TuringTest

so that's what goes outside the parentheses

- anonymous

7x(3x^2+2) Cant be right.

- TuringTest

what is
21x^2/7x=?
rewrite it in a way more comfortable to you if it looks strange

- anonymous

21/7 * x^2/x?

- TuringTest

yes...

- anonymous

Im pretty sure thats 3x man

- TuringTest

it is, so why did you write 3x^2 above?

- anonymous

Cause you said the other one wasnt right :/ What is the correct way to write it?

- TuringTest

here was your first answer
x(3x+2)
here was the next
21x^2/7x+14x/7x=7x(3x+2)
the answer here is correct, I just pointed out that the two expressions above are not equal, I think that may have confused you.
next you wrote
7x(3x^2+2)
so I think we started to get off track...
the answer is
21x^2+14x=7x(3x+2)
I just made a point about how you show your work, and what it means.

- anonymous

Ohhhhh :)

- TuringTest

write out the process as
21x^2+14x=7x(3x)+7x(2)=7x(3x+2)
that is really the best way to show factoring

- anonymous

Oh ok :D
Another one?

- TuringTest

factor
3t^3+9t^2

- TuringTest

sorry, so you know how to divide
t^3/t^2
???

- anonymous

x*x*x
----- right? It should jjust become x?
x*x

- anonymous

t, sorry

- TuringTest

you got it, so my question stands:
factor
3t^3+9t^2

- anonymous

GCF of 3 and 9 is 3.
GCF of t^2 and t^3 is t.
3t^3+9t^2=3t(3t^3/3t)+3t(9t^2/3t)=3t(t^3+3t^2)?

- TuringTest

check your answer and distribute to see if you get the original:
\[3t(t^3+3t^2)=3t(t^3)+3t(3t^2)=3t^4+9t^3\neq3t^3+9t^2\]you didn't divide the terms in the middle right
...but I think more importantly the GCF of t^3 and t^2 is t^2, not t
that is because each term can be evenly divided by t^2.

- anonymous

Hm

- anonymous

Oh right!

- TuringTest

what is the GCF of
t^3+t^5+t^7
??

- anonymous

Something to the power of something doesnt follow the rules of usual numbers, forgot that.
It should be t^5

- TuringTest

no it's t^3
and it does follow the rules of regular numbers if you think about it, otherwise it wouldn't be true!
look at
16,32,8
what is their GCF ?

- anonymous

8

- TuringTest

now rewrite those three numbers as powers of 2

- TuringTest

can you do that?

- anonymous

256,1024,64

- anonymous

?

- TuringTest

no I meant like
8=2^3
16=2^?
32=2^?

- anonymous

Oh!
8=2^3
16=2^4
32=2^5

- TuringTest

...and what is their GCF?
2^3
so the GCF of a set of variables is the highest common power to each term
-in this case 3
so if we have
y^4+y^7+y^9
the GCF is...?

- anonymous

y^4?

- TuringTest

right :)

- anonymous

:D

- TuringTest

good job :D
so now back to our question...
factor
3t^3+9t^2
what is the GCF?

- anonymous

3t^2?

- TuringTest

right
now can you factor it?

- anonymous

3t^3+9t^2=3t^2(3t^3/3t^2)+3t^2(9t^2/3t^2)=3t^2(t+3)?

- TuringTest

very nice!!!!!!

- anonymous

I got it!! :D

- TuringTest

you totally did :D
that is very cool!
so now lets see what's so great about factoring
say we have
f(t)=3t^3+9t^2
and we want to know the 'zeros' of the function
that means when the function touches the x-axis, i.e. when f(t)=0
so to answer this we must solve
0=3t^3+9t^2
how can we solve that?
by factoring...
0=3t^2(t+3)
now you can solve it quickly, any idea which fundamental rule of algebra tells us how?

- anonymous

Nope, sorry

- TuringTest

the rule is called the 'zero factor property'
it states that
if\[ab=0\]then either\[a=0\]or\[b=0\]or both
does this rule make logical sense to you?

- anonymous

Yeah

- TuringTest

so now look at our factored equation
\[0=ab=3t^2(t+3)\]that means that either\[3t^2=0\]or
\[t+3=0\]can you solve each of these equations?

- TuringTest

*either or both I should say

- anonymous

OH so 3t^2 represents a and t+3 represents b!

- TuringTest

exactly
that's why i said to learn the rules on this page
http://www.capitan.k12.nm.us/teachers/shearerk/basic_rules_of_algebra.htm
almost all of algebra is in there, though sometimes it is hidden

- anonymous

3t^2=0
t=0-3t/t
t=0-3
t=-3?

- TuringTest

no, it's more simple than that, remember the same rule we just used:
ab=0
then either a=0 or b=0 or both
3t^2=0
let 3=a t^2=b
we know that a=3 cannot be zero, because 3 is never zero, it's a constant
that leaves the possibility only of t^2=0
and the number number that times itself is zero is zero
so\[3t^2=0\to t=0\]

- anonymous

Ohhhh.. Cause 3x0x0=0?

- TuringTest

right
so t has to be zero...

- anonymous

Yeah

- TuringTest

what about the other possibility
t+3=0
???

- anonymous

How can that be possible when t is 0?

- TuringTest

there are two answers to every quadratic equation as you may recall me saying
this is cubic so it has 3 actually, we say that zero ocurrs twice in
3t^2=0 because it leads to two 0's as you showed: 3x0x0
we call that a 'multiplicity of 2'
so there will be multiple answers, the other is found by solving
t+3=0
remember that either a=0 or b=0 or BOTH
we don't know so we have to solve them all.

- anonymous

Oh right! This is a quadratic equation.
t=-3 on this one so its (0,-3)?

- TuringTest

like I said, cubic
read what I wrote above please about multiplicity...

- anonymous

Three answers

- anonymous

So these are cubic coordinates?

- anonymous

(0,0,-3)?

- TuringTest

not cubic coordinates (I don't know what that is exactly...),
it's just that we say that 3t^3+9t^2 has zeros
(0,3) - that means the graph of f(x)=3t^3+9t^2 hits zero there...
where 0 here has a 'multiplicity' of 2 (that means it occurs twice)
and 3 has a multiplicity of 1

- anonymous

So why isnt -3 involved?

- TuringTest

sorry typo, meant
(0,-3)*
and
-3 has multiplicity 1*

- TuringTest

good catch

- anonymous

Thanks

- TuringTest

so do you see what I mean?
the zero's of\[f(t)=3t^2+9t^2\]are found by factoring and setting to zero\[0=3t^2(t+3)\]then solving each possibility\[3t^2=0\to t=0\]\[t+3=0\to t=-3\]where we say that for
t=0 k=2
and for
t=-3 k=1
where k is the multiplicity

- anonymous

Right, the k is cause t is to the power of 3 there.

- TuringTest

for the part that had
3t^2=0 we had k=2 (because zero is the answer twice: 3x0x0)
for
t+3=0 we have k=1 because t is only to the first power and we only have one answer
If what you mean is that you noticed that adding up all the k's gave youu 3, the order of the cubic, then you have noticed what is called the Fundamental Theorem of Algebra:
"The sum of the multiplicities of the roots of a function is equal to the order of the polynomial"
\[k_0+k_{âˆ’3}=2+1=3\]

- TuringTest

- a very important theorem as the name implies...

- anonymous

I dont really get the theorem, please explain it.

- TuringTest

Basically for whatever the highest power variable you have in a polynomial, that is how many answers you have.
Note that they may not be all different answers, but the ones that occur more than once are counted as having a higher multiplicity, so if you add up the multiplicities (the k's) that's how many zeros the polynomial has.
for example
7x^5+3x^3+2x^2+5x+3=0
must have 5 answers, because it is 5th order
x^2+2x+2=0
must have 2 answers, because it is second order

- anonymous

So x^4+5x-4=0
must have 4 answers?

- TuringTest

exactly, though it may not be 4 different answers
for instance
x^4=0
has only the answer x=0, but that zero has a multiplicity of k=4 because
0x0x0x0=0
is how it must be...

- anonymous

Oh, now i get it

- TuringTest

good :)
do you want to try some more factoring problems?
or perhaps you should learn a little about exponents first?
or perhaps you are ready top get some rest....
which is it?

- anonymous

I wanna learn something new :D

- TuringTest

let's see if this is news to you:
simplify\[{x^{14}\over x^{12}}\]

- anonymous

x^2 lol

- TuringTest

good that saves a lot of time...

- TuringTest

simplify\[\sqrt[3]{x^{21}}\]

- anonymous

x^7?

- TuringTest

good :)
more time saved...

- anonymous

:D actually i didnt know that

- TuringTest

the genereal rule is\[\sqrt[b]{x^{a}}=x^{a/b}\]any radical can be wriitten as a fractional exponent, for instance\[\sqrt x=x^{1/2}\]so...
let's try some of that.

- anonymous

Yeah, i assumed that :)

- TuringTest

good guess, try to really rationalize it if you can...
since you seem to know the rules let's try a trickier one
simplify
\[\frac{\sqrt[3]{x^2}\sqrt[5]{x^3}}{\sqrt x}\]

- anonymous

x^2/3 * x^3/5
----------- Its basically this, right? If so, i just need to make the variables "suitable" to
x^1/2
be merged, right?

- TuringTest

right, just remember that\[x^ax^b=x^{a+b}\]and that\[\frac{x^a}{x^b}=x^{a-b}\]so you're gonna have and subtract to add those fractions, so they all need a common denominator.

- TuringTest

so you're gonna have to add and subtract those fractions*

- anonymous

Yeah, thats what i meant by making them suitable to merge

- TuringTest

I figured, but that's not a known term to me, just making sure

- anonymous

x^2/3 * x^3/5 x^10/15 * x^9/15 x^19/15
----------- = =--------------
x^1/2 x^1/2
Now i need to get 19/15 divisible by 2

- TuringTest

good so far :)

- anonymous

19/15 ------------38/30

- anonymous

x^38/30
-------- =
x^1/2
Actually, dont i need to get them divisible by 15 so i can get 2 as the denominator?

- TuringTest

what can you multiply the bottom by to get 1/2 over 30 ?

- anonymous

Oh!
x^38/30
-------- = 23/30? No, that cant be it..
x^15/30

- TuringTest

yes it can and is
but you have to leave the x of course...
x^(23/30)

- anonymous

Oh right lol

- TuringTest

so that was good, I won't test you on that
do you know how to FOIL ?

- anonymous

What?

- TuringTest

i.e. simplify
(a+b)(c+d)

- anonymous

abcd?

- TuringTest

FOIL=
First
Outer
Inner
Last|dw:1325979609902:dw|watch the arrows
they multiply the first in the brackets, the outer terms, the inner, and the last seperateyl

- TuringTest

it's just like distribution, but you have to do a and b seperate

- anonymous

I think i get it. give me another one and ill solve it fast

- TuringTest

It's okay if you don't solve it fast
actually remember that solving is not simplifying, here we have no = sign so we are simplifying:
(x+3)(x+2)

- anonymous

x^2+2x+3x+6?

- TuringTest

nice!
you can simplify the middle terms

- anonymous

Oh right,x^2+5x+6

- TuringTest

nicely done!
especially for never having FOILed before.
simplify
(2x+5)(3x-1)

- anonymous

5x+2x+15x-5?

- anonymous

6x*

- TuringTest

careful...
what's the first?
what's the outer?
the others are right.

- TuringTest

yes the first is 6x
the outer though...

- anonymous

-1?

- TuringTest

times what?

- anonymous

2x, maybe thats -2x..

- TuringTest

there ya go ;-)

- TuringTest

so it should be...?

- anonymous

6x+-2x+15x-5

- TuringTest

scratch the extra plus sign, but yes, now simplify...

- anonymous

29x-5?

- TuringTest

oh my mistake, you forgot that the first term would be squared
First=2x(3x)=6x^2
now what do you get?

- anonymous

6x^2-2x+15x-5?

- TuringTest

yes, and now simplify

- anonymous

4x^2+15x-5?

- TuringTest

like terms, x with x....
not x with x^2...

- anonymous

so 6x^2+13x-5?

- TuringTest

yes. good job.
now simplify
(2p-3)(5p-1)

- anonymous

10p^2-2p-15p-3? Simplified:10p^2-17p-3

- TuringTest

good, but watch the last term
negative times negative is...?

- anonymous

Oh right, positive.
So 10p^2-17p+3?

- anonymous

2r^2+2r*3t+t*r+3t^2 Simplified: 2r^2+2r^2+3t^2+3t^2

- TuringTest

stop before your simplification (which is wrong, sorry)
look at the first step:
2r^2+2r*3t+t*r+3t^2
^^^
should everything be positive?

- anonymous

Oh right..
2r^2+2r*-3t+t*r

- TuringTest

yes but you forgot the last term
you should also put parentheses around the negative terms

- anonymous

2r^2+2r*(-3t)+t*(-r)?

- TuringTest

lol it got deleted, what was the original problem please?

- anonymous

I cant remember :(

- TuringTest

ok another, because we started losing terms in the last one
(2y+x)(4y-3x)

- anonymous

8y+2y*3x+x*4y(-3x^2)

- TuringTest

there should be four terms
what is your first? write out the middle step

- TuringTest

it's close in many ways, but there are quite a few mistakes

- anonymous

Ill put them in prenthesis
(8y)+(2y*3x)+(x*4y)-(3x^2)

- TuringTest

better, but the first is
2y(4y) no?
and the outer is a positive times a negative as well

- anonymous

Oh, ill solve this one and then sleep :)
(8y^2)+(2y*-3x)+(x*4y)-(3x^2)

- TuringTest

nice, so to simplify this first multiply the coefficients...

- anonymous

(8y^2)+(8y^2*-3x^2)-(3x^2)

- TuringTest

not I think you're getting tired
(8y^2)+(2y*-3x)+(x*4y)-(3x^2)=8y^2-6xy+4xy-3x^2=?

- TuringTest

no*

- TuringTest

can you simplify the last step?

- anonymous

Yeah, i need to sleep.

- TuringTest

for future reference:\[(8y^2)+(2y*-3x)+(x*4y)-(3x^2)\]\[=8y^2-6xy+4xy-3x^2=8y^2-2xy-3y^2\]goodnight!

- anonymous

Good night! Thanks for all youve taught me :D See you tomorrow?

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