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  
  is a root of the polynomial, then the complex conjugate of  
  is also a root.
For example the polynomial  
  has a root  
  so according to the argument above  
  is also a root.
We can solve the equation  
  using the quadratic formula  
\[z=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]
\[a=1, \: b=2, \: c=2\]
) 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  
Suppose now that  
\[w, \: w^*\]
  are complex conjugates.
The sum and product of complex conjugates are real (If  

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