## Roots of Polyniomials With Real Coefficients

In general a polynomial of degree n hase n root, some of them possibly equal. The roots may be real or complex, but if the coefficients of the polynomial are real and a complex number
$z$
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
$z^2+2z+2=0$
$z=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$
(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
$ww^*=(x+iy)(x-iy)=x^2+ixy-ixy-i^2y^2=x^2+y^2$
.