How do you find a polynomial function of degree 3 with 5, i, -i as zeros?

1 Answer
CJ
May 19, 2015

The best way is to recognise that, if #x=5# is a root, then #x-5=0#, and ditto for the other two roots. So we have #x-5, x-i, x+i# all equalling zero. To find our polynomial, we just multiply the three terms together:

#(x-5)(x-i)(x+i)#
#=(x^2-ix-5x+5i)(x+i)#
#=x^3+ix^2-ix^2-(i^2)x-5x^2-5ix+5ix+5i^2#
which simplifies to
#x^3-5x^2+x-5#.

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