anonymous
  • anonymous
TuringTest, want to continue?
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
TuringTest
  • TuringTest
lol I like the attitude, I better eat though, give me a bit what time is it where you are?
anonymous
  • anonymous
9:45 pm, you? Then go eat :)
TuringTest
  • 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

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

anonymous
  • anonymous
Oh, ill use that!
TuringTest
  • TuringTest
see you in a bit
anonymous
  • anonymous
Alright
anonymous
  • anonymous
How do you use those fan messages Turing?
anonymous
  • anonymous
Go to the profile and click "write a fan message"
TuringTest
  • TuringTest
factor 21x^2+14x
anonymous
  • anonymous
GCF of 21 and 14=7 GCF of x and x^2 is x 21x^2/7x+14x/7x=x(3x+2)?
TuringTest
  • 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
  • anonymous
But i thought that if the 7 was on the LHS then it couldnt be on the RHS?
anonymous
  • anonymous
21x^2/7x+14x/7x=7x(3x+2)?
TuringTest
  • 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
  • anonymous
I dont know...
anonymous
  • anonymous
7(3x^2+2)?
TuringTest
  • TuringTest
you almost have it but you need to put the GCF outside the parentheses what is the GCF?
anonymous
  • anonymous
7x
TuringTest
  • TuringTest
so that's what goes outside the parentheses
anonymous
  • anonymous
7x(3x^2+2) Cant be right.
TuringTest
  • TuringTest
what is 21x^2/7x=? rewrite it in a way more comfortable to you if it looks strange
anonymous
  • anonymous
21/7 * x^2/x?
TuringTest
  • TuringTest
yes...
anonymous
  • anonymous
Im pretty sure thats 3x man
TuringTest
  • TuringTest
it is, so why did you write 3x^2 above?
anonymous
  • anonymous
Cause you said the other one wasnt right :/ What is the correct way to write it?
TuringTest
  • 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
  • anonymous
Ohhhhh :)
TuringTest
  • 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
  • anonymous
Oh ok :D Another one?
TuringTest
  • TuringTest
factor 3t^3+9t^2
TuringTest
  • TuringTest
sorry, so you know how to divide t^3/t^2 ???
anonymous
  • anonymous
x*x*x ----- right? It should jjust become x? x*x
anonymous
  • anonymous
t, sorry
TuringTest
  • TuringTest
you got it, so my question stands: factor 3t^3+9t^2
anonymous
  • 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
  • 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
  • anonymous
Hm
anonymous
  • anonymous
Oh right!
TuringTest
  • TuringTest
what is the GCF of t^3+t^5+t^7 ??
anonymous
  • anonymous
Something to the power of something doesnt follow the rules of usual numbers, forgot that. It should be t^5
TuringTest
  • 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
  • anonymous
8
TuringTest
  • TuringTest
now rewrite those three numbers as powers of 2
TuringTest
  • TuringTest
can you do that?
anonymous
  • anonymous
256,1024,64
anonymous
  • anonymous
?
TuringTest
  • TuringTest
no I meant like 8=2^3 16=2^? 32=2^?
anonymous
  • anonymous
Oh! 8=2^3 16=2^4 32=2^5
TuringTest
  • 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
  • anonymous
y^4?
TuringTest
  • TuringTest
right :)
anonymous
  • anonymous
:D
TuringTest
  • TuringTest
good job :D so now back to our question... factor 3t^3+9t^2 what is the GCF?
anonymous
  • anonymous
3t^2?
TuringTest
  • TuringTest
right now can you factor it?
anonymous
  • anonymous
3t^3+9t^2=3t^2(3t^3/3t^2)+3t^2(9t^2/3t^2)=3t^2(t+3)?
TuringTest
  • TuringTest
very nice!!!!!!
anonymous
  • anonymous
I got it!! :D
TuringTest
  • 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
  • anonymous
Nope, sorry
TuringTest
  • 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
  • anonymous
Yeah
TuringTest
  • 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
  • TuringTest
*either or both I should say
anonymous
  • anonymous
OH so 3t^2 represents a and t+3 represents b!
TuringTest
  • 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
  • anonymous
3t^2=0 t=0-3t/t t=0-3 t=-3?
TuringTest
  • 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
  • anonymous
Ohhhh.. Cause 3x0x0=0?
TuringTest
  • TuringTest
right so t has to be zero...
anonymous
  • anonymous
Yeah
TuringTest
  • TuringTest
what about the other possibility t+3=0 ???
anonymous
  • anonymous
How can that be possible when t is 0?
TuringTest
  • 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
  • anonymous
Oh right! This is a quadratic equation. t=-3 on this one so its (0,-3)?
TuringTest
  • TuringTest
like I said, cubic read what I wrote above please about multiplicity...
anonymous
  • anonymous
Three answers
anonymous
  • anonymous
So these are cubic coordinates?
anonymous
  • anonymous
(0,0,-3)?
TuringTest
  • 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
  • anonymous
So why isnt -3 involved?
TuringTest
  • TuringTest
sorry typo, meant (0,-3)* and -3 has multiplicity 1*
TuringTest
  • TuringTest
good catch
anonymous
  • anonymous
Thanks
TuringTest
  • 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
  • anonymous
Right, the k is cause t is to the power of 3 there.
TuringTest
  • 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" \[k_0+k_{−3}=2+1=3\]
TuringTest
  • TuringTest
- a very important theorem as the name implies...
anonymous
  • anonymous
I dont really get the theorem, please explain it.
TuringTest
  • 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
  • anonymous
So x^4+5x-4=0 must have 4 answers?
TuringTest
  • 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
  • anonymous
Oh, now i get it
TuringTest
  • 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
  • anonymous
I wanna learn something new :D
TuringTest
  • TuringTest
let's see if this is news to you: simplify\[{x^{14}\over x^{12}}\]
anonymous
  • anonymous
x^2 lol
TuringTest
  • TuringTest
good that saves a lot of time...
TuringTest
  • TuringTest
simplify\[\sqrt[3]{x^{21}}\]
anonymous
  • anonymous
x^7?
TuringTest
  • TuringTest
good :) more time saved...
anonymous
  • anonymous
:D actually i didnt know that
TuringTest
  • 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
  • anonymous
Yeah, i assumed that :)
TuringTest
  • 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
  • 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
  • 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
  • TuringTest
so you're gonna have to add and subtract those fractions*
anonymous
  • anonymous
Yeah, thats what i meant by making them suitable to merge
TuringTest
  • TuringTest
I figured, but that's not a known term to me, just making sure
anonymous
  • 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
  • TuringTest
good so far :)
anonymous
  • anonymous
19/15 ------------38/30
anonymous
  • 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
  • TuringTest
what can you multiply the bottom by to get 1/2 over 30 ?
anonymous
  • anonymous
Oh! x^38/30 -------- = 23/30? No, that cant be it.. x^15/30
TuringTest
  • TuringTest
yes it can and is but you have to leave the x of course... x^(23/30)
anonymous
  • anonymous
Oh right lol
TuringTest
  • TuringTest
so that was good, I won't test you on that do you know how to FOIL ?
anonymous
  • anonymous
What?
TuringTest
  • TuringTest
i.e. simplify (a+b)(c+d)
anonymous
  • anonymous
abcd?
TuringTest
  • 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
  • TuringTest
it's just like distribution, but you have to do a and b seperate
anonymous
  • anonymous
I think i get it. give me another one and ill solve it fast
TuringTest
  • 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
  • anonymous
x^2+2x+3x+6?
TuringTest
  • TuringTest
nice! you can simplify the middle terms
anonymous
  • anonymous
Oh right,x^2+5x+6
TuringTest
  • TuringTest
nicely done! especially for never having FOILed before. simplify (2x+5)(3x-1)
anonymous
  • anonymous
5x+2x+15x-5?
anonymous
  • anonymous
6x*
TuringTest
  • TuringTest
careful... what's the first? what's the outer? the others are right.
TuringTest
  • TuringTest
yes the first is 6x the outer though...
anonymous
  • anonymous
-1?
TuringTest
  • TuringTest
times what?
anonymous
  • anonymous
2x, maybe thats -2x..
TuringTest
  • TuringTest
there ya go ;-)
TuringTest
  • TuringTest
so it should be...?
anonymous
  • anonymous
6x+-2x+15x-5
TuringTest
  • TuringTest
scratch the extra plus sign, but yes, now simplify...
anonymous
  • anonymous
29x-5?
TuringTest
  • TuringTest
oh my mistake, you forgot that the first term would be squared First=2x(3x)=6x^2 now what do you get?
anonymous
  • anonymous
6x^2-2x+15x-5?
TuringTest
  • TuringTest
yes, and now simplify
anonymous
  • anonymous
4x^2+15x-5?
TuringTest
  • TuringTest
like terms, x with x.... not x with x^2...
anonymous
  • anonymous
so 6x^2+13x-5?
TuringTest
  • TuringTest
yes. good job. now simplify (2p-3)(5p-1)
anonymous
  • anonymous
10p^2-2p-15p-3? Simplified:10p^2-17p-3
TuringTest
  • TuringTest
good, but watch the last term negative times negative is...?
anonymous
  • anonymous
Oh right, positive. So 10p^2-17p+3?
anonymous
  • anonymous
2r^2+2r*3t+t*r+3t^2 Simplified: 2r^2+2r^2+3t^2+3t^2
TuringTest
  • 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
  • anonymous
Oh right.. 2r^2+2r*-3t+t*r
TuringTest
  • TuringTest
yes but you forgot the last term you should also put parentheses around the negative terms
anonymous
  • anonymous
2r^2+2r*(-3t)+t*(-r)?
TuringTest
  • TuringTest
lol it got deleted, what was the original problem please?
anonymous
  • anonymous
I cant remember :(
TuringTest
  • TuringTest
ok another, because we started losing terms in the last one (2y+x)(4y-3x)
anonymous
  • anonymous
8y+2y*3x+x*4y(-3x^2)
TuringTest
  • TuringTest
there should be four terms what is your first? write out the middle step
TuringTest
  • TuringTest
it's close in many ways, but there are quite a few mistakes
anonymous
  • anonymous
Ill put them in prenthesis (8y)+(2y*3x)+(x*4y)-(3x^2)
TuringTest
  • TuringTest
better, but the first is 2y(4y) no? and the outer is a positive times a negative as well
anonymous
  • anonymous
Oh, ill solve this one and then sleep :) (8y^2)+(2y*-3x)+(x*4y)-(3x^2)
TuringTest
  • TuringTest
nice, so to simplify this first multiply the coefficients...
anonymous
  • anonymous
(8y^2)+(8y^2*-3x^2)-(3x^2)
TuringTest
  • TuringTest
not I think you're getting tired (8y^2)+(2y*-3x)+(x*4y)-(3x^2)=8y^2-6xy+4xy-3x^2=?
TuringTest
  • TuringTest
no*
TuringTest
  • TuringTest
can you simplify the last step?
anonymous
  • anonymous
Yeah, i need to sleep.
TuringTest
  • 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
  • anonymous
Good night! Thanks for all youve taught me :D See you tomorrow?

Looking for something else?

Not the answer you are looking for? Search for more explanations.