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Find a polynomial equation with real coefficients that has the given roots.

5 and -7i

Write a polynomial equation with roots 5 and -7i

x^3 - __ x^2 + __ x -__ = 0

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  1. Don't know if they still teach that, but ....
    for any quadratic equation of the form
    ax^2 + bx + c = 0
    the sum of the roots is -b/a , and the product of the roots is c/a

    complex roots always appear as conjugate pairs, so they must be
    5-7i and 5+7i
    sum = 10
    product = (5-7u)(5+7i) = 25 - 49i^2 = 25+49 = 74

    equation must be
    x^2 - 10x + 74 = 0

    or .... if the roots are 5-7i and 5+7i
    then (x - (5-7i))(x - (5+7i)) = 0
    (x - 5 + 7i)(x - 5 - 7i) = 0
    x^2 - 5x - 7ix - 5x + 25 + 35i + 7ix - 35i - 49i^2 = 0
    x^2 - 10x + 25 - 49i^2 = 0 , but i^2 = -1

    x^2 - 10x + 74 = 0

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