A Delightful Solution (77) - Manipulating Polynomials and Roots of Unity

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  • Published on Jan 31, 2019
  • We obliterate the problem using the art of clever manipulations - what all mathematicians enjoy.
    Congratulations to Rishav Gupta, Kartik Sharma, Xu Chen Tan, Francesco Debortoli, Simon Armstrong, and The vast universe for successfully solving this math challenge question! Rishav Gupta was the first person to solve the question.
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Comments • 35

  • Nasir Khan
    Nasir Khan 2 months ago

    This just brilliant. Your videos always fascinated me and helped me to develop my ability to put on logical exertions in such a beautifully constructed problem.

  • SamRaxh' o_O
    SamRaxh' o_O 4 months ago

    How can 2020-(1010×2019) equal -170!?

  • Aarya Chaumal
    Aarya Chaumal 4 months ago

    How do I reach to u
    Like a message

    • Aarya Chaumal
      Aarya Chaumal 4 months ago

      @LetsSolveMathProblems f(x)=sin(sin(sin...70 sines... (x)
      Find seventh derivative of f(x) at x=0
      😋
      Also if possible give a representation for n nested sines of such a function
      All the best🤗

    • LetsSolveMathProblems
      LetsSolveMathProblems  4 months ago

      Please feel free to share the problem here on the comment section. =)

    • Aarya Chaumal
      Aarya Chaumal 4 months ago

      I have a nice que I would like u to make a video on😁
      Pls

  • Miyuki Umeki
    Miyuki Umeki 4 months ago

    how can I solve these insane questions by myself?

  • SeongKyun Jung
    SeongKyun Jung 4 months ago

    I failed this time, got to the polynomials but didn't think of comparing with x^2019+x^2018+...+1. I'll solve it next time tho! Great problem..

  • max power
    max power 4 months ago +1

    what tablet / input device do you use ?

  • Rahul Patil
    Rahul Patil 4 months ago

    Sir can u plz guide me how to integrate e^sinx

  • Debopam Sil
    Debopam Sil 4 months ago +2

    Oh goodness. I thought that omega was the complex cube root of 1 while solving the question. Just as I clicked on the video to see if my solution was correct or not, I discovered to my horror that omega is not what I thought. However, for those interested, my answer came out to be 1.

  • liam
    liam 4 months ago +1

    What software did u use to make this video?

  • Ben Burdick
    Ben Burdick 4 months ago

    Yeah, this one went completely over my head...

    • LetsSolveMathProblems
      LetsSolveMathProblems  4 months ago

      If there are any specific parts of the explanation you wish me to further elaborate on, please feel free to comment it here. I would love to help you fill in the missing pieces. =)

  • William Debdoot
    William Debdoot 4 months ago +2

    Sir, please solve some [JEE Advanced] mathematics problems (Indian Institute of Technology entrance exam).

  • G Dunken
    G Dunken 4 months ago

    Yes, we are done.

  • SUPRATIM SANTRA
    SUPRATIM SANTRA 4 months ago +1

    Sir @ beautiful... Where can i get such beautiful collection of problem?

  • fengsheng Qin
    fengsheng Qin 4 months ago +5

    I can't understand the "P(x)" rule ,why it equal to X^2020 -1 ? like P(x)=(x-1)(x-2)(x-4)=x*3-7x^2+14x-8 not x^3-1 ,pls explain why ,thanks!

    • PDV Cubing
      PDV Cubing 2 months ago

      There’s a theorem in algebra where, if a polynomial p(x) has leading coefficient a and roots r1, r2, r3, ..., rN, then p(x) can be expressed as:
      p(x) = a(x-r1)(x-r2)•••(x-rN)
      The nth roots of unity are all the complex numbers that, when raised to the nth power, are equal to 1. Algebraically, they’re the solutions to the equation:
      x^n = 1
      By rearranging this equation, we can clearly see that they are also the roots of the polynomial
      p(x) = x^n - 1
      When n = 2020,
      p(x) = x^2020 - 1
      1 is a trivial solution, and all the complex solutions are powers of ω = e^(2iπ/2020), which can be written in the form ω^k, k = 1,2,3,...,2019. Using the rule from the first paragraph, we can rewrite x^2020 - 1 in terms of its roots,
      p(x) = (x-1)(x-ω)(x-ω^2)•••(x-ω^2019)
      This is sort of simplified so please let me
      know if I got anything wrong!

    • antonofka9018
      antonofka9018 4 months ago

      @LetsSolveMathProblems I don't understand what you did at 8:25

    • antonofka9018
      antonofka9018 4 months ago

      Watch this video usclip.net/video/8GPy_UMV-08/video.html

    • LetsSolveMathProblems
      LetsSolveMathProblems  4 months ago +4

      I believe you may not have learned about the roots of unity, which is a prerequisite to thoroughly understanding this video and, in fact, the problem. Here goes a brief introduction.
      It can be proven that, given any complex number z and positive integer n, there are exactly n different complex numbers such that each is an nth root of z. That is, there are 3 complex cube roots of 1; there are 2 complex square roots of 2019+2020i; etc.
      The complex numbers that are roots of 1 (the unity) are named roots of unity. It can be proven that nth roots of unity are 1, w, w^2, ... , w^(n-1) where w = e^(2pi*i/n). In our case, n = 2020.
      Consider a polynomial x^3-1. The zeroes of this polynomial are complex numbers such that x^3-1 = 0, or x^3 = 1, that is, the cube roots of unity. Thus, x^3-1 = (x-1)(x-w)(x-w^2), where w = e^(2pi*i/3). Note that 2 and 4 are NOT roots of unity, so the example you provided does not work.
      I hope this helps. If you wish me to further elaborate on some of the above points or provide you with resources for learning more about roots of unity, I am willing to. =)

    • Omar
      Omar 4 months ago

      Correct. and as you see it was given:
      ω=e^(2πi/2020) which corresponds with the solution of the equaution x^2020=1⇒ x=e^(2kπi/2020) (in polar form) ∀k 0≤k≤2019 which implies
      x=(e^(πi/1010))^k which is equivalent to (as you see) ω,ω²,ω³...,ω^2019.

  • UbuntuLinux
    UbuntuLinux 4 months ago +10

    I understand
    .
    .
    .
    .
    .
    .
    nothing...

    • LetsSolveMathProblems
      LetsSolveMathProblems  4 months ago +2

      If there are any specific parts of the explanation you wish me to further elaborate on, please feel free to comment it here. I would love to help you fill in the missing pieces. =)

  • Supanat Sukpiriyakul
    Supanat Sukpiriyakul 4 months ago +1

    What is​ that big pi

    • Johannes H
      Johannes H 4 months ago +1

      If you watch the video, you see it

    • Zυвi
      Zυвi 4 months ago +2

      Same as sigma notation but you multiply instead of adding

    • Joe Potillor
      Joe Potillor 4 months ago +2

      Product notation

  • nizar aberqi
    nizar aberqi 4 months ago

    Great