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TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1lol I like the attitude, I better eat though, give me a bit what time is it where you are?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.09:45 pm, you? Then go eat :)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1alright, just so you know it's easy to send messages via fan message now, so use that instead of a question to hail me

Nodata
 2 years ago
Best ResponseYou've already chosen the best response.0How do you use those fan messages Turing?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Go to the profile and click "write a fan message"

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0GCF of 21 and 14=7 GCF of x and x^2 is x 21x^2/7x+14x/7x=x(3x+2)?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1almost 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

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0But i thought that if the 7 was on the LHS then it couldnt be on the RHS?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.021x^2/7x+14x/7x=7x(3x+2)?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1but 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)=?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1you almost have it but you need to put the GCF outside the parentheses what is the GCF?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1so that's what goes outside the parentheses

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.07x(3x^2+2) Cant be right.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1what is 21x^2/7x=? rewrite it in a way more comfortable to you if it looks strange

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Im pretty sure thats 3x man

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1it is, so why did you write 3x^2 above?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Cause you said the other one wasnt right :/ What is the correct way to write it?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1here 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.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1write out the process as 21x^2+14x=7x(3x)+7x(2)=7x(3x+2) that is really the best way to show factoring

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1sorry, so you know how to divide t^3/t^2 ???

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0x*x*x  right? It should jjust become x? x*x

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1you got it, so my question stands: factor 3t^3+9t^2

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0GCF 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
 2 years ago
Best ResponseYou've already chosen the best response.1check 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.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1what is the GCF of t^3+t^5+t^7 ??

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Something to the power of something doesnt follow the rules of usual numbers, forgot that. It should be t^5

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1no 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 ?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1now rewrite those three numbers as powers of 2

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1no I meant like 8=2^3 16=2^? 32=2^?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Oh! 8=2^3 16=2^4 32=2^5

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1...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...?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1good job :D so now back to our question... factor 3t^3+9t^2 what is the GCF?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1right now can you factor it?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.03t^3+9t^2=3t^2(3t^3/3t^2)+3t^2(9t^2/3t^2)=3t^2(t+3)?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1you 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 xaxis, 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?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1the 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?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1so 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
 2 years ago
Best ResponseYou've already chosen the best response.1*either or both I should say

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0OH so 3t^2 represents a and t+3 represents b!

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1exactly 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

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.03t^2=0 t=03t/t t=03 t=3?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1no, 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\[3t^2=0\to t=0\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1right so t has to be zero...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1what about the other possibility t+3=0 ???

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0How can that be possible when t is 0?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1there 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.

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Oh right! This is a quadratic equation. t=3 on this one so its (0,3)?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1like I said, cubic read what I wrote above please about multiplicity...

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0So these are cubic coordinates?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1not 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

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0So why isnt 3 involved?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1sorry typo, meant (0,3)* and 3 has multiplicity 1*

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1so 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

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Right, the k is cause t is to the power of 3 there.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1for 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
 2 years ago
Best ResponseYou've already chosen the best response.1 a very important theorem as the name implies...

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0I dont really get the theorem, please explain it.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1Basically 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

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0So x^4+5x4=0 must have 4 answers?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1exactly, 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...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1good :) 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?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0I wanna learn something new :D

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1let's see if this is news to you: simplify\[{x^{14}\over x^{12}}\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1good that saves a lot of time...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1simplify\[\sqrt[3]{x^{21}}\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1good :) more time saved...

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0:D actually i didnt know that

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1the 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.

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah, i assumed that :)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1good guess, try to really rationalize it if you can... since you seem to know the rules let's try a trickier one simplify \[\frac{\sqrt[3]{x^2}\sqrt[5]{x^3}}{\sqrt x}\]

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0x^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
 2 years ago
Best ResponseYou've already chosen the best response.1right, just remember that\[x^ax^b=x^{a+b}\]and that\[\frac{x^a}{x^b}=x^{ab}\]so you're gonna have and subtract to add those fractions, so they all need a common denominator.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1so you're gonna have to add and subtract those fractions*

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah, thats what i meant by making them suitable to merge

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1I figured, but that's not a known term to me, just making sure

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0x^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

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.019/15 38/30

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0x^38/30  = x^1/2 Actually, dont i need to get them divisible by 15 so i can get 2 as the denominator?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1what can you multiply the bottom by to get 1/2 over 30 ?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Oh! x^38/30  = 23/30? No, that cant be it.. x^15/30

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1yes it can and is but you have to leave the x of course... x^(23/30)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1so that was good, I won't test you on that do you know how to FOIL ?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1i.e. simplify (a+b)(c+d)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1FOIL= First Outer Inner Lastdw:1325979609902:dwwatch the arrows they multiply the first in the brackets, the outer terms, the inner, and the last seperateyl

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1it's just like distribution, but you have to do a and b seperate

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0I think i get it. give me another one and ill solve it fast

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1It'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)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1nice! you can simplify the middle terms

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1nicely done! especially for never having FOILed before. simplify (2x+5)(3x1)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1careful... what's the first? what's the outer? the others are right.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1yes the first is 6x the outer though...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1scratch the extra plus sign, but yes, now simplify...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1oh my mistake, you forgot that the first term would be squared First=2x(3x)=6x^2 now what do you get?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1yes, and now simplify

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1like terms, x with x.... not x with x^2...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1yes. good job. now simplify (2p3)(5p1)

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.010p^22p15p3? Simplified:10p^217p3

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1good, but watch the last term negative times negative is...?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Oh right, positive. So 10p^217p+3?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.02r^2+2r*3t+t*r+3t^2 Simplified: 2r^2+2r^2+3t^2+3t^2

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1stop before your simplification (which is wrong, sorry) look at the first step: 2r^2+2r*3t+t*r+3t^2 ^^^ should everything be positive?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Oh right.. 2r^2+2r*3t+t*r

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1yes but you forgot the last term you should also put parentheses around the negative terms

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1lol it got deleted, what was the original problem please?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1ok another, because we started losing terms in the last one (2y+x)(4y3x)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1there should be four terms what is your first? write out the middle step

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1it's close in many ways, but there are quite a few mistakes

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Ill put them in prenthesis (8y)+(2y*3x)+(x*4y)(3x^2)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1better, but the first is 2y(4y) no? and the outer is a positive times a negative as well

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Oh, ill solve this one and then sleep :) (8y^2)+(2y*3x)+(x*4y)(3x^2)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1nice, so to simplify this first multiply the coefficients...

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0(8y^2)+(8y^2*3x^2)(3x^2)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1not I think you're getting tired (8y^2)+(2y*3x)+(x*4y)(3x^2)=8y^26xy+4xy3x^2=?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1can you simplify the last step?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1for future reference:\[(8y^2)+(2y*3x)+(x*4y)(3x^2)\]\[=8y^26xy+4xy3x^2=8y^22xy3y^2\]goodnight!

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Good night! Thanks for all youve taught me :D See you tomorrow?
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