Wikipedia:Reference desk/Archives/Mathematics/2011 May 6

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May 6

Factoring equations into single polynomials

I can't for the life of me figure this one out: (x^2+2)^3 - (3x^3-4)^2

I managed to get the first half as x^6+8x^2+8 but I realized I was stuck on the second half becuse I can't use difference of squares on it. Now what? --Thebackofmymind (talk) 11:13, 6 May 2011 (UTC)

Why don't you use the formula (a+b)^3 and (a-b)^2? See polynomial identities.-Shahab (talk) 11:56, 6 May 2011 (UTC)

I don't really understand the question, but it could be that you are just misunderstanding the word "factoring". Factoring is like when you are given x^2 + 3x + 2 and you have to express it as (x + 1)(x + 2). What you seem to be doing is multiplying polynomial expressions together, which the reverse process and much easier. Where does the factoring come in? 86.181.202.126 (talk) 12:03, 6 May 2011 (UTC)

Ah, you're both right. Thank you for the help! I need more sleep.

I seemed to have mixed it up with another problem I was trying to factor, (x-2)^5. If you kind sirs would be able to help me with this one too, it would be nice. --Thebackofmymind (talk) 12:31, 6 May 2011 (UTC)

(x-2)^5 is already factorised - it has five factors, each of which is x-2. I think you may have been asked to expand this expression, in which case our article on binomial expansion may help you. Gandalf61 (talk) 12:40, 6 May 2011 (UTC)

So far what I tried to do was write out

(x-2)(x-2)(x-2)(x-2)(x-2)

(x-2)(x^2-4x+4)(x^2-4x+4)

(x-2)(x^4-8x^3+12x^2-32x+16)

x^5-10x^4+28x^3-56x^2+80x+32

But the answer key says I'm doing it wrong. I'll take a look at the article Gandalf, thanks.--Thebackofmymind (talk) 12:31, 6 May 2011 (UTC)

Your method is okay, but there are errors in your arithmetic - for example, the last term should be -32 not +32. The binomial theorem provides a short-cut method that avoids some of the long-hand multiplication. Gandalf61 (talk) 12:55, 6 May 2011 (UTC)

Yeah, you're right. Thanks for showing me that article! So I took the original problem I had and put it into the equations Shahab provided and got this:

(x² + 2)³ - (3x³ - 4)²

[(x²)³ + 2³ + 3(2)(x²)(x²+2)] -[(3x³)² - 2(3x³)(4) + 4²

[x⁶ + 8 + 6x²(x² + 2)] - [3x⁶ - 24x³ + 16]

-2x⁶ + 6x⁴ + 24x³ + 12x² -8

But I'm still wrong! I supposedly had everything right except the first term, which is supposedly -8x⁶...but how am I supposed to get -8 for a coefficient when I only have x⁶ and -3x⁶ to play with? --Thebackofmymind (talk) 21:53, 6 May 2011 (UTC)

(3x^3)^2=9x^6 not 3x^6. You have to square the 3 too. --Tango (talk) 23:30, 6 May 2011 (UTC)

Oh wow. That was a noobish mistake. Thank you for pointing it out! --Thebackofmymind (talk) 17:44, 7 May 2011 (UTC)

http://www.wolframalpha.com/input/?i=(x^2%2B2)^3+-+(3x^3-4)^2 Bo Jacoby (talk) 09:16, 7 May 2011 (UTC).

Thanks for the link! I've been looking for a free program like that, but I'm surprised at what else you can put in! --Thebackofmymind (talk) 17:44, 7 May 2011 (UTC)

GCE "A" level maths after the switch from GCEs to GCSEs

I have a copy in front of me of a revision book for GCE "O" level maths (Letts Key Facts Traditional and Modern Mathematics by K. Ahmad MA) and it covers calculus and quite a lot of other densly-written stuff.

I also have another revision book for GCSE maths (Mathematics Revision Notes For GCSE by P. Jenkins) and it does not mention calculus at all, and the treatment of for example set theory is much simpler and less detailed. This is despite it being designed for those aiming at the higher grades.

My question is - what happened to the gap between GCSE and GCE "A" level? Was GCE "A" level maths dumbed down because the kids had to spend time learning stuff they would have previously learnt at "O" level? Thanks 92.28.243.102 (talk) 13:28, 6 May 2011 (UTC)

You are correct. When I did "O" level maths in the 70s, differentiation and integration of xⁿ and polynomials was covered at the top end of the "O" level course. When my sons did GCE "A" level maths a few years ago the same topics had migrated into the bottom end of the "A" level course - in the Core 1 module, if I remember correctly, so typically covered in the first term of the "AS" year. I am not sure what might have dropped off the top end of the "A" level syllabus to compensate. On the other hand, the range of modules on offer makes the modern "A" level course broader - my sons did stuff in the Decision and Statistics modules that I didn't cover until University. Gandalf61 (talk) 13:47, 6 May 2011 (UTC)

It might have depended on the particular board. When I did my O and A levels in the early 80's I first did calculus at the start of A levels. Two thing which seems particularly absent from the A level syllabus these days is matrices and complex numbers.--Salix (talk): 14:17, 6 May 2011 (UTC)

Calculus is an unfortunately scary-sounding name for a beautiful and quite literally invaluable area of mathematics: without the tools it provides our world would be a significantly less advanced one, all manner of modern technology relies on calculus and the areas which develop from it. I think it comes up too often in modern tv shows and films as the throwback term for a 'scary' sounding course which all the students struggle with because it's so boring and hard. Quite the opposite in fact; if you do persevere and put the effort into grasping the underlying concepts, it can be a richly rewarding area to study - in addition, in my opinion it's often the future science/mathematics/economics/etc student's first taste of what begins to branch into the sort of mathematics which is used at research levels and for other scientific professions. I'm just finishing my final year (doing maths, of course!) at university so I was in your position recently.

I think the current curriculum is very poorly designed personally, in that it teaches you the right thing to do is ignore the underlying concepts and just learn how to implement a method you don't understand because it's quicker and a more direct route to the exam marks. (http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html may interest you - though it's American, almost all of the points made apply to the UK too) As such, students aren't really prepared to actually think as mathematics requires you to, only as exams require you to, and I suspect that is precisely why the gap between A-level and GCSE-level had to be narrowed. Of course, exam boards and the media love to report that there are "more 'A*'s than ever this year", implying that the country is becoming precisely 3 or 4% smarter every year (with any luck by the turn of the next century we will have evolved into some sort of floating brain) and our education system is unrivalled, rather than the incredible possibility our exams are getting easier.

You should also remember that they now offer further mathematics as an A-level option: I'm not sure whether or not this was available when O-levels existed or not but I wouldn't expect so. So anyway, perhaps it is simply the case that the material has in fact increased in volume, it is simply that it has not increased by enough to spread across two A-levels rather than one and so the material inevitably gets thinned out all the way back to GCSEs. Of course, the easier the GCSEs are the more students are likely to continue doing maths up to A-level, which looks much better for the government than an abundance of 'soft' subjects like textiles. (Forgive me, but I'm extremely cynical about mathematics education in the UK right up until university where the government has less remit to interfere and the subject can be taught properly, rather than as an exercise in box-ticking.)

Incidentally, I covered both matrices and complex numbers at A-level: I think the latter was at the very start of my further mathematics A-level, but I was under the impression matrices were on the regular A-level: perhaps I'm misremembering. Either way, it was merely calculating the determinant of a 3x3 matrix (here is a formula: insert numbers here.) so not really worth batting an eyelid for. The very final topic I myself covered in further mathematics A-level was a small spattering of group theory, but I believe that it's only the OCR exam board which group theory is covered on. It's a shame because it's an amazing subject, and probably the first truly abstract concept a maths student will ever encounter. What a tragedy that all the most amazing and beautiful mathematics is squandered only for the people who study it to university and beyond, eh? Anyway, there's my two cents (pennies?), for what it's worth. Tasterpapier (talk) 14:21, 6 May 2011 (UTC)

I would also heartily recommend Lockhart's Lament, which funnily enough I discovered for the first time on this very reference desk a few years ago. http://www.maa.org/devlin/LockhartsLament.pdf - it's exceedingly good, well worth a quick flip through at least if you have the time, it makes me think a little of Richard Dawson if he directed his concerns to the teaching of maths rather than evolution. It's semi-related, but very interesting. Tasterpapier (talk) 15:07, 6 May 2011 (UTC)

Yes it has been massively and deliberately dumbed down, for that and other reasons. The article which covers it is that on grade inflation, especially the section on A-levels.--JohnBlackburne^words_deeds 18:19, 6 May 2011 (UTC)

The conflation of grade inflation and the "dumming down" of courses should not be made. Employers and universities are perfectly capable of judging the relative success of a candidate, and now demand three As when they previously wanted three BBBs. That aside, there has been some adjustment. Gandalf is correct to say calculus is taught in the first module of A level; parts of the old A level maths course are in the Further Maths course, like matrices and complex numbers. (Transformations are taught at GCSE without matrices.) This has partly been a result of efforts to move "the bits people don't get" out the practically compulsory GCSE and into an optional, if widely taken, A level. The second issue has been that of the balance between "pure" (now synonymous with the term "core") and applied mathematics (now statistics, mechanics and decision maths), which partly took the form of a move from four to six modules. From what I can see, it would be possible to cover the "old" topics completely if five, rather than four, of the six maths modules were pure (essentially the first 'pure' further maths module), clearly at the expense of applied maths. Grandiose (me, talk, contribs) 20:00, 6 May 2011 (UTC)

To me 'grade inflation' is the formal name for 'dumbing down' which is what the original poster asked about. If you want to separate them maybe grade inflation is when, as is happening with degree results, the grades of students increase over time so ever more are getting firsts. Dumbing down is where the desired grade, e.g. 'C' for O-level/GCSE, is being reached by ever greater numbers because the exams have got easier as has been objectively measured (see e.g. the references in grade inflation). But really it is difficult to separate them, as they really are two different ways of thinking of the same phenomena: exams getting easier so more people pass them and/or those doing them get better grades.--JohnBlackburne^words_deeds 20:24, 6 May 2011 (UTC)

"Was GCE "A" level maths dumbed down because the kids had to spend time learning stuff they would have previously learnt at "O" level?" implies the OP wanted to know whether the course itself had changed, rather than expectations of the result of it. (This takes a meaning of "grade inflation" that implies an equal level of skill results in a higher grade.) They are quite separate phenomena, although the do come along together.Grandiose (me, talk, contribs) 20:32, 6 May 2011 (UTC)

When I did O level maths in 1958 or so (JMB) I recall no calculus, and think that this came in the subsequent A level. So if I'm right, for this board at least any inclusion of calculus at O level came later. I've kept a lot of old exam papers, but not from as long ago as this, so can't be sure.→86.132.165.117 (talk) 19:45, 7 May 2011 (UTC)

Yes, the old "Syllabus A" had separate Algebra, Arithmetic and Geometry papers with no calculus (but formal proofs in Euclidean Geometry), whilst the O-level "Syllabus B" included calculus in the 1960s. On the introduction of GCSE courses, it was decided that calculus was more appropriate to an A-level course. (Basic methods for differentiating and integration poylnomials could be taught to a bright 7-year-old, (I learnt them by chance when I was 12) but it's the application that's important, and proper understanding is more likely at a later age.) Also, Further Maths A-level has been available since the 1950s. Dbfirs 08:32, 8 May 2011 (UTC)

Antisymmetric Matrix

Hi, If I have an antisymmetric matrix R, how might I show that there are real 3D vectors x and y and some real number z such that Rx=zy and Ry=-zx? I know that it is true intuitively because it is like a rotation matrix... Thanks!131.111.222.12 (talk) 14:39, 6 May 2011 (UTC)

I'm assuming R is 3 by 3 otherwise the question doesn't make sense. A 3 by 3 antisymmetric matrix has determinant 0 so there is a vector v with Rv = 0. So one easy, but probably not intended solution is x=y=v, z=0. The other eigenvalues of R are pure imaginary (assuming R is not 0). Any vector v in the corresponding eigenspaces will have R²v=-qv where q is a positive real (in fact the sum of the squares of the entries of R above the diagonal). If v is an eigenvector of one imaginary eigenvalue, then its conjugate is an eigenvector of the other imaginary eigenvalue, so taking x as the real part, it will satisfy R²x=-qx. Take z=√q and y=z^-1Rx for the other solution. Probably not the most elegant solution.--RDBury (talk) 15:41, 6 May 2011 (UTC)

Thanks, RDBury. (And yes, your assumption is true.) —Preceding unsigned comment added by 131.111.222.12 (talk) 17:38, 6 May 2011 (UTC)

How about this: R² is real symmetric, so its eigenvalues are real and have real eigenvectors. Furthermore $\textstyle {\textrm {tr}}(R^{2})=-\sum _{i,j}R_{ij}^{2}\leq 0$. If the trace is zero then R = 0; otherwise R² has at least one negative eigenvalue, so call it −z² and let x be a corresponding real eigenvector and y = z^-1Rx. This works in any number of dimensions (if you drop "3D" from the question). -- BenRG (talk) 03:08, 10 May 2011 (UTC)

Probability (2 player game)

Hi, If 2 players engage in a series of games in which player 1 has prob p of winning, and P2 has prob q=1-p of winning, and the game ends when one player wins exactly 3 more games than the other, what is the prob that, say, P1 wins the overall game? I have come up with different ways of tackling this, but they are all quite long-winded and inelegant. I am wondering if anyone has a better suggestion...

The ways I approach this were:

1. to use the law of total prob and consider the P(P1 wins and P1 wins by the end of 1st 5 games)+P(P1 wins if tie by the end of 5th game)+P(P1 wins and P2 wins by the end of 1st 5 games) (the last is clearly 0 just included for completeness) but then working out the 1st 2 probs takes a lot of work...

2. try to sum the probs that P1 wins after chunks of 5 games (infinite sum). but this is also tedious...

Thanks! —Preceding unsigned comment added by 131.111.222.12 (talk) 20:58, 6 May 2011 (UTC)

I added to the title, since there's already another question with the same vague title. StuRat (talk) 21:21, 6 May 2011 (UTC)

Well, I think the area of maths you need is the theory of random walks (the amount player 1 is winning by takes a random walk, with each step being +1 with probability p and -1 with probability q). I don't know much about them, though, so can't really help you. --Tango (talk) 23:48, 6 May 2011 (UTC)

Don't hold me to this because I haven't fully worked it through, but you may be able to do it by defining f(n) as the probability of the first player winning given a current win/loss difference of n. Then f(n) = p*f(n+1) + q*f(n-1). Then you solve this recurrence relation, plug in the appropriate "initial" conditions which would be something like f(3) = 1, f(-3) = 0, and then you want f(0). 86.181.204.0 (talk) 01:45, 7 May 2011 (UTC)

Hi guys, thanks for your suggestions! I have worked through the prob using the random walk model, as suggested by Tango and explicitly stated by 86.181.204.0. —Preceding unsigned comment added by 131.111.222.12 (talk) 01:49, 7 May 2011 (UTC)