Roots of Polyniomials With Real Coefficients
\[z\]
is a root of the polynomial, then the complex conjugate of \[z\]
written \[z*\]
is also a root.For example the polynomial
\[z^2+2z+2\]
has a root \[z=-1+i\]
so according to the argument above \[z^*=-1-i\]
is also a root.We can solve the equation
\[z^2+2z+2=0\]
using the quadratic formula \[z=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]
(with \[a=1, \: b=2, \: c=2\]
) to give \[z=\frac{-2 \pm \sqrt{2^2-4 \times 1 \times 2}}{2 \times 1}=\frac{-2 \pm \sqrt{=4}}{2}= -1 \pm i\]
.We can write the polynomal above as
\[(z-(1+i))(z-(1-i))\]
.Suppose now that
\[w, \: w^*\]
are complex conjugates.\[(z-w)(z-w^*)=z^2-zw^*-zw+ww*=z^2-z(w+w^*)+ww^*\]
.The sum and product of complex conjugates are real (If
\[w=x+iy\]
then \[w^*=x-iy\]
sp \[w+w^*=2x\]
and \[ww^*=(x+iy)(x-iy)=x^2+ixy-ixy-i^2y^2=x^2+y^2\]
. 
