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March 29, 2017

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f(x)=x^4+ax^3+bx^2+cx+d and has real coefficients. If two of the roots are 4−i and 2+3i, what is the value of a+b+c+d?

  • Math (Algebra) - ,

    complex roots always come in conjugate pairs. So,

    f(x) = (x-(4-i))(x-(4+i))(x-(2+3i))(x-(2-3i))
    = ((x-4)^2 + 1^2)((x-2)^2 + 3^2)
    = (x^2-8x+17)(x^2-4x+13)
    = x^4 - 12x^3 + 62x^2 - 172x + 221
    ...

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