Re: HSC 2013 4U Marathon
Another question on polys/complex numbers:
Suppose
is a root of the polynomial.
Let
such that
Then by substituting this into our polynomial and letting it equal to 0 (since we are assuming it's a root) and by using De Moivre's Theorem, we obtain:
If we plotted these points on an Argand Diagram,
would be on the positive real axis given that
.
The leading term would lay in the first two quadrants, unless
, in which case it would lay on the negative real axis.
All terms in between would lay in the first two quadrants such that their imaginary components are positive.
By vector addition, since all points have
, we can never arrive at 0, and hence
Therefore we have a contradiction - our assumption is wrong. Hence we can deduce that
has no roots
where
Note by a symmetrical argument we can justify it holds true when all the points obey the condition
.