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March 30, 2017

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The function f(x)=x^4−15(x^3)+81(x^2)−201x+182 has four complex roots, one of which is 3−2i. What is the sum of all real and imaginary coefficients of these roots?

Details and assumptions
i is the imaginary unit, where i2=−1.

  • Maths - ,

    one other complex root must be 3+2i
    So, (x-(3-2i))(x-(3+2i)) are factors of f(x)

    That is, (x-3)^2+4 = (x^2-6x+13) divides f(x)

    f(x) = (x^2-6x+13)(x^2-9x+14)
    = (x^2-6x+13)(x-2)(x-7)

    So, the roots are
    2,7,3-2i,3+2i
    You can add 'em up

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