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In mathematics, the complex conjugate root theorem states that if ''P'' is a polynomial in one variable with real coefficients, and ''a'' + ''bi'' is a root of ''P'' with ''a'' and ''b'' real numbers, then its complex conjugate ''a'' − ''bi'' is also a root of ''P''.〔 Preview available at (Google books )〕 It follows from this (and the fundamental theorem of algebra), that if the degree of a real polynomial is odd, it must have at least one real root. That fact can also be proven by using the intermediate value theorem. == Examples and consequences == * The polynomial ''x''2 + 1 = 0 has roots ±''i''. * Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue. * The polynomial :: :has roots :: :and thus can be factored as :: :In computing the product of the last two factors, the imaginary parts cancel, and we get :: :The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complex conjugate root theorem」の詳細全文を読む スポンサード リンク
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