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  
\[z^2+2z+2=0\]
  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  
\[ww^*=(x+iy)(x-iy)=x^2+ixy-ixy-i^2y^2=x^2+y^2\]
.

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