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Math Help - Polynomial equation

  1. #1
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    Polynomial equation

    Determine the polynomial equation in standard form that has the following roots:

    -1 (of order 2), and (2+2sqrt 3) and (2-2sqrt 3)

    please show all work!
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  2. #2
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    Hello, meli3000!

    Determine the polynomial equation in standard form that has the following roots:

    . . -1\:\text{(of order 2)},\;\,2+2\sqrt{3},\;\;2-2\sqrt{3}

    If the polynomial P(x) has root -1\text{ (order 2)}, then (x+1)^2 is a factor of P(x).

    If 2 + 2\sqrt{3} is a root, then x - (2 + 2\sqrt{3}) is a factor.

    If 2 - 2\sqrt{3} is a root, then x - (2 - 2\sqrt{3}) is a factor.


    Hence: . P(x) \;=\;(x+1)^2(x - [2+2\sqrt{3}])(x - [2-2\sqrt{3}]) \;=\;0

    And we have: . (x^2 + 2x + 1)(x^2-4x - 8) \;=\;0

    . . Therefore: . \boxed{x^4 - 2x^3 - 15x^2 - 20x - 8 \;=\;0}

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