To prove that -----  x³ + p x + q = 0 if ---?

To prove that ----- x³ + p x + q = 0 if ---

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  • Anonymous
    4 months ago
    Favorite Answer

    ...........................

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  • Indica
    Lv 7
    4 months ago

      ..................................................................

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  • 4 months ago

    The answer is as follows:

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  • Plug that in for x and see what you get.

    a = -q/2

    b = sqrt((1/4) * q^2 + (1/27) * p^3)

    ((a + b)^(1/3) + (a - b)^(1/3))^3 + p * ((a + b)^(1/3) + (a - b)^(1/3)) + q

    Let's work on each term.  First, x^3

    (a + b)^(3/3) + 3 * (a + b)^(2/3) * (a - b)^(1/3) + 3 * (a + b)^(1/3) * (a - b)^(2/3) + (a - b)^(3/3) =>

    a + b + 3 * (a + b)^(1/3) * (a + b)^(1/3) * (a - b)^(1/3) + 3 * (a + b)^(1/3) * (a - b)^(1/3) * (a - b)^(1/3) + a - b =>

    2a + 3 * (a + b)^(1/3) * (a - b)^(1/3) * ((a + b)^(1/3) + (a - b)^(1/3)) =>

    2a + 3 * ((a + b) * (a - b))^(1/3) * x =>

    2a + 3 * (a^2 - b^2)^(1/3) * x =>

    2a + 3 * ((1/4) * q^2 - (1/4) * q^2 - (1/27) * p^3)^(1/3) * x =>

    2a + 3 * ((-1/27) * p^3)^(1/3) * x =>

    2a + 3 * (-1/3) * p^(3/3) * x =>

    2a - px

    Now we have:

    2a - px + px + q =>

    2a + q =>

    2 * (-q/2) + q =>

    -q + q =>

    0

    0 = 0

    Done.

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