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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Polynomial function theorems for zeros ： ウィキペディア英語版 Polynomial function theorems for zeros Polynomial function theorems for zeros are a set of theorems aiming to find (or determine the nature of) the polynomial remainder theorem: * Factor theorem * Descartes' rule of signs * Gauss–Lucas theorem * Rational zeros theorem * Bounds on zeros theorem also known as the ''boundedness theorem'' * Intermediate value theorem * Complex conjugate root theorem * Properties of polynomial roots ==Background== A polynomial function is a function of the form :$p(x)\; =\; a\_n\; x^n\; +\; a\_\; x^\; +\; \backslash dotsb\; +\; a\_2\; x^2\; +\; a\_1\; x\; +\; a\_0$, where $a\_i\backslash ,\; (i\; =\; 0,\; 1,\; 2,\; \backslash dotsc,\; n)$ are complex numbers and $a\_n\; \backslash ne\; 0$. If $p(z)\; =\; a\_n\; z^n\; +\; a\_\; z^\; +\; \backslash dotsb\; +\; a\_2\; z^2\; +\; a\_1\; z\; +\; a\_0\; =\; 0$, then $z$ is called a ''zero'' of $p(x)$. If $z$ is real, then $z$ is a ''real zero'' of $p(x)$; if $z$ is imaginary, the $z$ is a ''complex zero'' of $p(x)$, although complex zeros include both real and imaginary zeros. The fundamental theorem of algebra states that every polynomial function of degree $n\; \backslash ge\; 1$ has at least one complex zero. It follows that every polynomial function of degree $n\; \backslash ge\; 1$ has exactly $n$complex zeros, not necessarily distinct. * If the degree of the polynomial function is 1, i.e., $p(x)\; =\; a\_1\; x\; +\; a\_0$, then its (only) zero is $\backslash frac$. * If the degree of the polynomial function is 2, i.e., $p(x)\; =\; a\_2\; x^2\; +\; a\_1\; x\; +\; a\_0$, then its two zeros (not necessarily distinct) are $\backslash frac$. A degree one polynomial is also known as a linear function, whereas a degree two polynomial is also known as a quadratic function and its two zeros are merely a direct result of the quadratic formula. However, difficulty rises when the degree of the polynomial, ''n'', is higher than 2. There is a cubic formula for a cubic function (a degree three polynomial) and there is a quartic formula for a quartic function (a degree four polynomial), but they are very complicated. There is no general formula for a polynomial function of degree 5 or higher (see Abel–Ruffini theorem). 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「Polynomial function theorems for zeros」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.