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  • Today's curve.

    From Richard Hachel@21:1/5 to All on Fri Mar 28 12:33:37 2025
    Hello, math lovers.

    We will now present today's curve.

    Let the function f(x)=x⁴-2x²+8 be given.

    First question (4 points):
    1. Give the coordinates (x,y) of three points through which this curve
    f(x) passes.

    Second question (4 points):
    2. Give the equation of the function g(x) with point symmetry $(0,y₀).

    Third question (6 points):
    3. Give the two complex roots of f(x).

    Fourth question (6 points):
    4. Verify that the answers are correct by including them in f(x).

    R.H.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Jim Burns@21:1/5 to Richard Hachel on Fri Mar 28 12:10:45 2025
    On 3/28/2025 8:33 AM, Richard Hachel wrote:

    Let the function f(x)=x⁴-2x²+8 be given.

    Define '⋅'
    ⟨0,1⟩⋅⟨0,1⟩ = ⟨-1,0⟩
    ⟨1,0⟩⋅⟨0,1⟩ = ⟨0,1⟩
    ⟨0,1⟩⋅⟨1,0⟩ = ⟨0,1⟩
    ⟨1,0⟩⋅⟨1,0⟩ = ⟨1,0⟩

    f(⟨x,y⟩) = ⟨x,y⟩⁴-2⟨x,y⟩²+8⟨1,0⟩

    It follows that
    f(⟨x,y⟩) = ⟨0,0⟩ ⇔
    ⟨x,y⟩ ∈ {⟨a,b⟩,⟨-a,b⟩,⟨a,-b⟩,⟨-a,-b⟩}

    f(⟨x,y⟩) = ⟨x-a,y-b⟩⋅⟨x+a,y-b⟩⋅⟨x-a,y+b⟩⋅⟨x+a,y+b⟩

    a = 2³ᐟ⁴cos(½tan⁻¹(√7))
    b = 2³ᐟ⁴sin(½tan⁻¹(√7))

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Richard Hachel@21:1/5 to All on Fri Mar 28 18:38:35 2025
    Le 28/03/2025 à 17:10, Jim Burns a écrit :
    On 3/28/2025 8:33 AM, Richard Hachel wrote:

    Let the function f(x)=x⁴-2x²+8 be given.

    Define '⋅'
    ⟨0,1⟩⋅⟨0,1⟩ = ⟨-1,0⟩
    ⟨1,0⟩⋅⟨0,1⟩ = ⟨0,1⟩
    ⟨0,1⟩⋅⟨1,0⟩ = ⟨0,1⟩
    ⟨1,0⟩⋅⟨1,0⟩ = ⟨1,0⟩

    f(⟨x,y⟩) = ⟨x,y⟩⁴-2⟨x,y⟩²+8⟨1,0⟩

    It follows that
    f(⟨x,y⟩) = ⟨0,0⟩ ⇔
    ⟨x,y⟩ ∈ {⟨a,b⟩,⟨-a,b⟩,⟨a,-b⟩,⟨-a,-b⟩}

    f(⟨x,y⟩) = ⟨x-a,y-b⟩⋅⟨x+a,y-b⟩⋅⟨x-a,y+b⟩⋅⟨x+a,y+b⟩

    a = 2³ᐟ⁴cos(½tan⁻¹(√7))
    b = 2³ᐟ⁴sin(½tan⁻¹(√7))

    This is indeed what mathematicians say, but it's neither true nor
    beautiful.

    In good mathematical logic, there are only two roots, both of the complex
    root type.

    Let x'=2i and x"=-2i.

    The way mathematicians deal with all this is as ridiculous as it is
    shameful.

    Almost criminal.

    R.H.

    --- SoupGate-Win32 v1.05
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