CVPL3

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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-a_{i+1})\,

=(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\,

Main Page : Complex Variables : Polynomials

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