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I was working on question 5 and I got stuck. I can use rational root theorem to show that it has no rational roots. However, I do not know how to show that it has no real roots. Can someone guide me?

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## Comments

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TopNewestPlease help with the second solution to E3(a).

I don't understand why we're taking \(n=3k, 3k+1, 3k+2\), or why getting a zero after substituting \(x=\omega\) in \(x^{2n}+x^{n}+1\) means divisibility by \(x^{2}+x+1\). How would this process come to mind?

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\(x^{2}+x+1 = (x - \omega)(x - \omega^{2})\).

Mow let us take \(f(x) = x^{2n}+x^{n}+1\).

So if we have \(f(\omega) = 0\) and \(f(\omega^{2})=0\) ,then we can write by the factor theorem \[(x - \omega)(x - \omega^{2}) | f(x)\]

that is

\[x^{2}+x+1 | f(x)\]

Now the divisibility of \(f(x)\) is totally dependent on the value of \(n\) modulo 3 and hence we have to consider 3 different cases as given by the book.

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Okay, I understand why the substitution works, but I still don't get why taking those values of

nwork. Please help.Log in to reply

How many of the problems do you expect us to be able to do and in how much time per problem? I could do all of them except for #37, but problems like #5, #3, #4, #18 took a few days of thought.

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The time taken will greatly depend on your ability level, and familiarity with proofs. I intended for people to spend some time thinking about these problems, which is why this stretches across 2 weeks.

You should give yourself a pat on the back for solving these problems. They are not easy, and are above the level of what is normally covered in high school.

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You have a very good start. By the way, what troubles you in ques 37.??

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It's just that I am not very familiar with functional equations.

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Firstly,

a function is not a polynomial! So you can't determine it's coefficient or anything of the sort. So we have a new toolkit:Guess the functions. Normally we guess forlinear and constantfunctions. Guessing these functions provides us with a structure to solve the problem. Gut feel for functional equations is that only linear and constant functions will satisfy, but for some more complex ones, there might be higher powers.Substitute the variables with something else. Remember, you want to cancel stuff out, so if there are \( f \) on both sides of the equation, say \( f(a) + \text{blah} = f(b) \), then set try to manipulate the variables such that \( a=b \).But this problem does not require any of the machinery listed above, it is a \( polynomial \) functional equation, meaning that we can still use our toolkit for polynomials. But step \( 1 \) comes in really handy.

Next, notice that \( x^2 + x + 1 \), the complicated thing in this question, looks extremely like roots of unity. Once we notice

roots of unity, analysing thecomplex zeroessort of transforms this problem from a polynomial functional equation one to one which manipulates complex numbers.Then once we get some roots, (through playing around and trying to force out a contradiction at all steps if possible \( \Rightarrow \) this actually suggests

extremal principle) then we pull out the roots that we already have, (in this case writing \( f(x) = (x^2+1)^m g(x) \) ) at this step what we basically do is tosubstitute everything back into the original equationto solve for \(g(x) \) andremembering our constraintsat the same time.Log in to reply

Functional Equations Wiki pages?

Can you add these comments to theSelect "Write a summary", and then copy-paste your text into it (with minor edits). Thanks!

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How does one think of a the method given in Example #6 (on page 250) himself? Is there any alternative method? Please help.

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Read Useful Lemma, which is the next installation in the series.

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Is there any alternative or easier way to solve 11 in the book other than the factorizing method given in the book? I must say that the questions are brain-wracking and thus this one made me give up factorizing :(. I must also include that few of the questions asked by Calvin to solve had an application of symmetric polynomial functions too, thus he shouldn't have included them! I would like to know methods to simplify eqautions like 11. I quite didnt know how to proceed to prove #4 and #5 though I'd got all of the rest right. I would like to get some exposure to some olympiad proofs as I'd actually learnt form the answers of these questions ( 4& 5)! Thanks :)

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The first thing you might notice is that the powers in the equation are all

even. That suggests us to apply a substitution \(x=z^2\) to simplify things. Calling the resulting expression \(P(x)\) we have: \[P(x)=x^4+4x^3-10x^2+4x+1=0.\] Now to factorize this, we get help from theRational Root Theorem, which suggests us to try \(\pm 1\). We see that \(P(1)=0\), hence \((x-1)\) is a factor of \(P(x)\) by theRemainder-Factor Theorem. Now rearrange terms and factorize: \[P(x)=x^4-x^3+5x^3-5x^2-5x^2+5x-x+1\] \[=x^3(x-1)+5x^2(x-1)-5x(x-1)-1(x-1)\] \[=(x-1)(x^3+5x^2-5x-1)\] \[=(x-1)\cdot Q(x) ~~~[\text{say}]\] Note that \(Q(x)\) is also monic and the constant term is \(1\). So it's easy to applyRational Root Theoremagain. Since \(Q(1)=0\), \((x-1)\) is a factor of it. We again factorize: \[Q(x)=x^3-x^2+6x^2-6x+x-1=x^2(x-1)+6x(x-1)+1(x-1)=(x-1)(x^2+6x+1)\] Therefore: \[P(x)=(x-1)\cdot Q(x)=(x-1)(x-1)(x^2+6x+1)=(x-1)^2(x^2+6x+1)=0\] Now \(x=1\implies z=\pm\sqrt{x}=\pm 1\) is a root and bash the other factor with Quadratic formula.Log in to reply

Huh, you can also do it by noticing that the coefficients are symmetric with respect to the middle term and so you can pull it apart into the sum of two squares which share a common factor.

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Ah! Thanks! :)

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Sure!

First, let \[y=z^2\] then we have \[((y+1)^2)^2-(4y)^2=0\] factoring as a difference of squares we have \[(y^2+6y+1)(y^2-2y+1)=0\] then we can find the roots of the two quadratic factors to be \[-3\pm 2\sqrt{2}, 1\] so the solutions are \[z=\pm\sqrt{-3\pm2\sqrt{2}}, \pm{1}\] Sorry if those aren't simplified.

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Hi! Are you talking about problem no.11? Not example, Problem!

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## 11 on page 255, yes.

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\[ z^8 + 4z^6 - 10z^4 + 4z^2 + 1 = y^4 + 4y^3+ 6y^2 + 4 y + 1 - 16 y^2 = (y+1)^4 - 16y^2 \]

It wasn't obvious how you arrived at the conclusion, and adding the intermediate steps would help.

Very nice observation!

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Good question. That was one reason why I chose 11, because there is an 'easier' way to see the factorization.

5 is actually hard. I was surprised it was posted so early in the set.

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Yes. Indeed 5 is hard and I actually had to learn from its solutions :(. Could you tell me what would the follow-ups for this course be?

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What does #3 ask me to find?

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Remember to state what you are having trouble with. Which phrase do you not understand?

Let \(x_1, x_2 \) be the 2 roots of \( x^2 - 6x + 1 = 0 \).

Let \( A_n = x_1 ^n + x_2 ^ n \) for ever non-negative integer \(n\).

1) Show that \(A_n\) is an integer.

2) Show that \( 5 \) does not divide \(A_n\).

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Ok, thank you very much

I just don't know what is the problem asking for

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How to convert cubic's of the form \(x^3+bx^2+cx+d\) into \(x^3+px+q\) in order to apply the solving method given in the book, The book says it can be done by translation, What does that mean?

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Let the roots of the first cubic be \(\alpha, \beta,\) and \(\gamma\). By Vieta's theorem, we can see that \(\alpha + \beta + \gamma = -b\).

Translation means shifting functions. If we translate the function horizontally by \(k\), the roots become \(\alpha + k, \beta + k\) and \(\gamma + k\). Since the coefficient of \(x^2\) in the second cubic is \(0\), the sum of its roots is also \(0\). Try to find a value of \(k\) for which the sum of the new roots become \(0\).

Once that is done, and the new roots are obtained, by Vieta's again, we can get value of \(p\) and \(q\) in terms of \(b, c\) and \(d\).

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If we shift \(x\) by \(k\), it would change the coefficient of \(x^{3}\), wouldn't it?

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In this case, \(x^3 + bx^2 + cx + d\) would become \((x - k)^3 + b(x - k)^2 + c(x-k) +d\). Note that the term of \(x^3\) appears only once, and its coefficient is still \(1\).

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@Samuraiwarm Tsunayoshi even if the coefficient would have changed, we could have still divided by that coefficient to get a monic cubic

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Hi Ishan,

Yes...just as @Pranshu Gaba said you must first translate the polynomial by \(k\) units.and then since we must eliminate the \(x^{2}\) part we must find the value of k for which the co-efficient of \(x^{2}\) is 0.But in this case since the polynomial is monic the co-efficient of \(x^{3}\) will remain unchanged inspite of the transformation...try it out to get the idea :)

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Hi Eddie,

I feel that even if a polynomial is not monic, translating it in any direction does not change the leading coefficient. This is because, essentially we are not changing the leading term when we translate the function. What do you think?

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I'm curious to know how others solved #3.

Engel's solution involves taking

modulo 5and I'm not sure how that could come to mind. I used induction instead.Log in to reply

Hi Siladitya,

I think that one of the motivations behind using the modulo 5 was to eliminate the \(6S_{n-1}\) part to get \(s_n = s_{n-1}-s_{n-2}\) looks relatively better to infer a solution from.NOw if we can show that this sequence recurs after a value without arriving at 0 then we have already proved that for all n the sequence is not divisible by 5.

If you have solved it in a different way using induction than feel free to post your solution after the due date so that others can learn from it.

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Thanks Eddie!

I believe seeing other people's solutions after solving, or failing to solve, a problem might be insightful. Also, we might understand shortcomings in our own solutions. However, there seems to be no designated place for discussing the solutions to (and not difficulties with) problems in Engel. Where and when do we post our solutions and discuss them?

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What I learnt page.

Actually Calvin Sir instructed this sprint to be about 2 weeks long and hence if we post our solutions now many will not try the problems on their own.But one thing that you \(\textbf{can}\) do right now is make a generalization of the idea that you used to solve the problem and post it in form of a note!!Also make sure that you post a link to the note in theIf you need help with Latex then I can help.

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I didn't play around with the problem after I solved it, so I really don't know in which way to generalize. Should I talk about a monic, integer-coefficient polynomial of

ndegrees; change the divisor from 5 to a broader set of numbers (primes??!); work with other forms that appear in Vieta's formulae\(((x_1x_2)^{n} + (x_2x_3)^{n} + (x_3x_1)^{n})\)? It'd be nice if there was some clarification!Log in to reply

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Where did you learn your math from- Eddie?

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After I'd looked through some of the problems, I still don't know how to start, can anyone here tell me what should I do?

Example #13:

Should I expand the equation?

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What do you expect to find after expanding the equation? What do you notice about the given equation that might let you make progress on the problem? After expanding, what is your plan?

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I came to this and I think I am near to the solution, so now what? Use the roots of unity? And what is the value of \(a\) and \(b\)?

\(x^4-(a^2+b^2)x+a^2b^2=0\)

Edited:

\(x^4-(a^2+b^2)x^2+a^2b^2=0\)

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Is this ok?

\(x^4-(a^2+b^2)x^2+a^2b^2=y^2-(a^2+b^2)y+a^2b^2\)

By using Vieta's formula,

Sum of roots = \(a^2+b^2\)

Product of roots = \(a^2b^2\)

So the roots are \(a^2\) and \(b^2\)

Am I right?

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\((x-a)^4\)

\(=[(x-a)^2]^2\)

\(=(x^2-a^2)^2\)

\(=x^4-2a^2x^2+a^4\)

Then I do the same to others

\((x-b)^4=x^4-2b^2x^2+b^4\)

\((a-b)^4=a^4-2a^2b^2+b^4\)

\(x^4-2a^2x^2+a^4+x^4-2b^2x^2+b^4=a^4-2a^2b^2+b^4\)

\(2x^4-2a^2x^2-2b^2x^2=-2a^2b^2\)

\(x^4-(a^2+b^2)x^2+a^2b^2=0\)

Have I done anything wrong here?

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\((x-a)^2\not=x^2-a^2\)

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Thanks

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We want to factorize the polynomial \( f(x) = (x-a)^4 + (x-b)^4 - (a-b)^4 \).

Hint:What can we say about \(f(a)\) and \(f(b) \)?Log in to reply

By the remainder-factor Theorem,

\(a\) and \(b\) are the roots of \(f(x)\)

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Extension on #3

Show why \((\sqrt{2}+1)^n\) is very close to even integers for large integer \(n\).

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I guess it's something to do with this thing's binomial expansion and then cancelling all the surds to obtain a multiple of 2. I know that I am not fully right

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Does anyone wanna do this or should I explain a solution?

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Comment deleted Jun 27, 2014

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n'.Log in to reply

@Nathan Ramesh !

Read: "for large \( n \)." I will post a solution later today if nobody tries it - have some patience,Log in to reply

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When you say "try not to refer to solutions" on the https://brilliant.org/profile/calvin-8u8hog/sets/2-week-sprint-through-polynomials/, does that mean not referring to the solutions to the examples in the text (p. 245-250)?

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You are referencing:

I'm asking you to work on the problems listed on page 254-258. Solutions are given at the end of the chapter and you should not be referring to them.

With regards to

You should be reading through the solutions to the examples.

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@Calvin Lin Generally ,when solving Olympiad problems, after how many hours of trying should one look at the solutions?

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When you read solutions, don't simply see how it's done. An important aspect (which is often ignored), is to dissect and analyze them. Ask yourself:

1. Do I know enough to come up with this solution? If yes, why didn't I? If no, what do I need to know?

2. Is the solution motivated by some principle? What aspects of the question gave it away?

3. If I saw a similar (or even the exact same) problem in 6 months, can I solve it within an hour?

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doing stuff(not just sitting idle)! If you are doing USAMO, I'd say do stuff for at least 1-1.5 hours. Same for IMO.Log in to reply

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I haven't encountered roots of unity before. I think the material in the book is too insufficient to understand for first-timers like me. Can somebody guide me to a good resource for roots of unity? Thanks.

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Roots of Unity is also a pretty good resource.

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I read through the notes but I don't understand how to change something into exponential form, can anyone tell me how?

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Hi Ananay!

If you are comfortable with basic trigonometry and complex numbers, you should be able to graph them on the complex plane given a unit circle. Given this, you should look at the AoPSWiki - this link is probably the best short fundamental explanation of roots of unity I have seen on the internet. Also, here is a nice blog post on AoPS expanding on roots of unity.

Lastly, if you are looking for a more visual approach, read this. Hope that helps!

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Comment deleted Jun 24, 2014

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Be sure to check out the khan academy videos on the roots of unity!!!There Sal makes an excellent job to explain them for beginners...... Also check out this video by Patrick JMT Click here

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I'll admit they are a little confusing. If you're not one to memorize formulas, you can easily derive or find any root of unity by just drawing the unit circle and placing \(n\) equidistant points along the circumference (for the \(n\)th roots of unity). You can use basic trig from there. :D

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The issue is that some students who are trying to learn roots of unity without De Moivre's Theorem will have trouble doing this. I think @Ananay Agarwal is looking for something geometrical to get an intuition about it.

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So Is this note still active? May I ask a doubt?

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Can someone please explain the example problem solution for this article - I'm having trouble following it.

Thanks

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Sure! I commit to helping anyone in these topics to the maximum possible.

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@Mursalin Habib @Sreejato Bhattacharya @Shaan Vaidya @anup navin @Sudeep Salgia @Anqi Li @Dinesh Chavan @Trevor B. @Eddie The Head @Sharky Kesa @Pranjal Prashant @Pranshu Gaba @Finn Hulse @Mardokay Mosazghi @Zong Zhang @Samuraiwarm Tsunayoshi @David Lee @Ahaan Rungta @Krishna Ar @Jubayer Nirjhor @Vagish Jha @Nathan Ramesh @José Marín Guzmán @Aayush Mani

Thanks for volunteering to be mentors. When the members have any questions about the material, they will post their question here. Please check in when you can, and help to clear out doubts. If this post become too long / unwieldly, I will come up with an alternative.

If any of you would like to post relevant notes and problems, you can state the links in a comment and I'd review them. (This includes posts made in the past)

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Comment deleted Jun 24, 2014

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This could be a bit off-topic, but since this topic would be covered later here, can anyone explain or suggest any good e-reosurce to learn cyclic and symmetric factorization? I believe these along with roots of unity are used to simplify lots of complex equations! thnx!

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You should start a new comment for every question you have! To answer your question, here are some resources I can think of:

Also, I recommend taking small symmetric and cyclic polynomials and multiplying them and seeing how the expanded polynomials relate to the factors, etc. and just getting some experience. Hope that helps!

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@Ahaan Rungta i am not really good at symmetric and cyclic factorization but this really helped.

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You should make other posts like combinatorics, number theory, geometry, and another stuffs separately. To be honest, I'm completely suck at this kind of algebra and I haven't done this ever. I think I'll be just a student here. T__T

Hopefully, I'll come around and help at combinatorics and number theory. ^__^

Edit: I am completely confused what this topic is, so I just called it "this kind of algebra" . XD

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First of all, it's pretty obvious that the polynomials sprint is just a starter, so there will be other projects. I like how Calvin is taking these subjects once at a time, making things very organized.

Second, this is not abstract algebra.

Abstract algebrais a totally different field of math (pun intended)! This is basic polynomial algebra, and considering you are Level 5, there is no reason that you should not be able to help. This thread is for Levels 2-3 so you are free to spectate but you should probably let the level 2-3s be the students.Third, even I have my weak subjects. For example, I am better in algebra and number theory than I am in combinatorics. So don't feel bad if you are slightly less active in some of these sprints. Good to have you on board! :)

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@Suyeon Khim @Silas Hundt @Peter Taylor @David Mattingly @Calvin Lin @Arron Kau and all other staff members

When a topic is deleted it actually should disappear, this long list makes my head hurtLog in to reply