Question

TuringTest, want to continue?

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

Oh, ill use that!

Answer 5

see you in a bit

Answer 6

Alright

Answer 7

How do you use those fan messages Turing?

Answer 8

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

Answer 9

factor
21x^2+14x

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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)=?

Answer 15

I dont know...

Answer 16

7(3x^2+2)?

Answer 17

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

Answer 18

7x

Answer 19

so that's what goes outside the parentheses

Answer 20

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

Answer 21

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

Answer 22

21/7 * x^2/x?

Answer 23

yes...

Answer 24

Im pretty sure thats 3x man

Answer 25

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

Answer 26

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

Answer 27

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.

Answer 28

Ohhhhh :)

Answer 29

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

Answer 30

Oh ok :D
Another one?

Answer 31

factor
3t^3+9t^2

Answer 32

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

Answer 33

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

Answer 34

t, sorry

Answer 35

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

Answer 36

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)?

Answer 37

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.

Answer 38

Hm

Answer 39

Oh right!

Answer 40

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

Answer 41

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

Answer 42

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 ?

Answer 43

8

Answer 44

now rewrite those three numbers as powers of 2

Answer 45

can you do that?

Answer 46

256,1024,64

Answer 47

?

Answer 48

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

Answer 49

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

Answer 50

...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...?

Answer 51

y^4?

Answer 52

right :)

Answer 53

:D

Answer 54

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

Answer 55

3t^2?

Answer 56

right
now can you factor it?

Answer 57

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

Answer 58

very nice!!!!!!

Answer 59

I got it!! :D

Answer 60

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?

Answer 61

Nope, sorry

Answer 62

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?

Answer 63

Yeah

Answer 64

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?

Answer 65

*either or both I should say

Answer 66

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

Answer 67

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

Answer 68

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

Answer 69

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\[3t^2=0\to t=0\]

Answer 70

Ohhhh.. Cause 3x0x0=0?

Answer 71

right
so t has to be zero...

Answer 72

Yeah

Answer 73

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

Answer 74

How can that be possible when t is 0?

Answer 75

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.

Answer 76

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

Answer 77

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

Answer 78

Three answers

Answer 79

So these are cubic coordinates?

Answer 80

(0,0,-3)?

Answer 81

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

Answer 82

So why isnt -3 involved?

Answer 83

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

Answer 84

good catch

Answer 85

Thanks

Answer 86

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

Answer 87

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

Answer 88

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"
\[k_0+k_{−3}=2+1=3\]

Answer 89

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

Answer 90

I dont really get the theorem, please explain it.

Answer 91

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

Answer 92

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

Answer 93

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...

Answer 94

Oh, now i get it

Answer 95

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?

Answer 96

I wanna learn something new :D

Answer 97

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

Answer 98

x^2 lol