How to write a quartic equation with integral coefficient such that two of its roots are 8i and -2i?

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How to write a quartic equation with integral coefficient such that two of its roots are 8i and -2i?

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Tags: Quartic Equation, Roots

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One Response to 'How to write a quartic equation with integral coefficient such that two of its roots are 8i and -2i?'

1. Rich J - August 31st, 2008 at 7:46 pm

   for any polynomial with real coefficients, if it has a complex root z, then it’s complex conjugate z’ is also a root.

   so if 8i and -2i are to be roots, -8i and 2i must also be.

   you then get a polynomial of the form: (x-8i)(x+8i)(x+2i)(x-2i), up to a multiplicative constant.

   now
   (x-8i)(x+8i) = x^2 + 64
   and
   (x+2i)(x-2i) = x^2 + 4

   so your polynomial is (x^2 + 64) (x^2 + 4)
   expand if needed.
   the coefficients will be integers, so you don’t have to multiply by a constant; if the coefficients had been rational, you could have multiplied by a proper integer to get integral coefs.