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

1 Answer
May 19, 2015

The best way is to recognise that, if x=5x=5 is a root, then x-5=0x5=0, and ditto for the other two roots. So we have x-5, x-i, x+ix5,xi,x+i all equalling zero. To find our polynomial, we just multiply the three terms together:

(x-5)(x-i)(x+i)(x5)(xi)(x+i)
=(x^2-ix-5x+5i)(x+i)=(x2ix5x+5i)(x+i)
=x^3+ix^2-ix^2-(i^2)x-5x^2-5ix+5ix+5i^2=x3+ix2ix2(i2)x5x25ix+5ix+5i2
which simplifies to
x^3-5x^2+x-5x35x2+x5.