Precalculus

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Suppose that a polynomial function of degree 4 with rational coefficients has i and (-3 + square root of 3)as zeros find the other zeros

  • Precalculus -

    There must be four roots total and complex numbers have conjugates that are roots. One other root is therefore
    -i .
    If -3 + sqrt3, is a root, the another root is -3 - sqrt3, as a consequence of the +/-sqrt(b^2 - 4ac) in the quadratic equation.

    The polynomial must be a multiple of
    (x^2 +1)(x +3 -sqrt3)(x +3 +sqrt3) = 0
    (x^2+1)[(x+3)^2 -3] = 0
    (x^2+1)(x^2 +6x +6) = 0
    x^4 +6x^3 +7x^2 +6 = 0

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