## 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 \[ww^*=(x+iy)(x-iy)=x^2+ixy-ixy-i^2y^2=x^2+y^2\]

.