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

  • Published on Jan 31, 2019
  • We obliterate the problem using the art of clever manipulations - what all mathematicians enjoy.
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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😁

  • 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 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

    • 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