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Show that a polynomial with real coefficients can always be expressed as a product of linear and quadratic factors with real coefficients.

Imaginary roots of polynomials with real coefficients occur in pairs, so if one such pair was (z-a_{i})(z-a_{{i+1}})\, in the factorization a_{0}(z-a_{1})(z-a_{2})\cdot \cdot \cdot (z-a_{n})\, then those two factors can be combined to make a factor with all real coefficients.


=(z-a_{i})(z-{\bar  {a_{i}}})\,

=z^{2}-z(a_{i}+{\bar  {a_{i}}})+a_{i}{\bar  {a_{i}}}\,

=z^{2}-2z{\mathrm  {Re}}\,a_{i}+|a_{i}|^{2}\,

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