9.5: Aplicación a polinomios reales
- Page ID
- 118476
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Si\(p\) es un polinomio en\(\mathbb{R}[x]\), se deduce del Teorema Fundamental del Álgebra que sí tiene raíces, pero pueden ser complejas. Si tiene raíces complejas, deben ocurrir en pares conjugados complejos.
TEOREMA 9.46. Vamos\(p \in \mathbb{R}[x]\). Dejar\(\alpha\) ser una raíz de\(p\). Entonces así es\(\bar{\alpha}\).
PRUEBA. Vamos\(p(x)=\sum_{k=0}^{N} a_{k} x^{k}\). Entonces\[p(\alpha)=\sum_{k=0}^{N} a_{k} \alpha^{k}=0,\] Así que\[p(\bar{\alpha})=\sum_{k=0}^{N} a_{k} \bar{\alpha}^{k}=\overline{p(\alpha)}=0 .\] vamos\(\alpha=a+i b\). \[\begin{aligned} (x-\alpha)(x-\bar{\alpha}) &=(x-(a+i b))(x-(a-i b)) \\ &=x^{2}-2 a x+a^{2}+b^{2} \\ &=(x-a)^{2}+b^{2} . \end{aligned}\]Entonces, aplicando el Teorema Fundamental del Álgebra al polinomio real\(p\), primero factorizamos las raíces reales, y por cada par de raíces conjugadas complejas obtenemos un factor como en (9.47). Así conseguimos:
TEOREMA 9.48. Dejar\(p \in \mathbb{R}[x]\) ser un polinomio de grado\(N\). Entonces se\(p\) puede factorizar en un producto de factores lineales\(\left(x-c_{k}\right)\) y factores cuadráticos\(\left(\left(x-a_{k}\right)^{2}+b_{k}^{2}\right)\):\[p(x)=c\left(\prod_{k=1}^{N_{1}}\left(x-c_{k}\right)\right)\left(\prod_{j=1}^{N_{2}}\left(\left(x-a_{j}\right)^{2}+b_{j}^{2}\right)\right)\] para algunos números reales (no necesariamente distintos)\(c, c_{j}, a_{j}, b_{j}\). Tenemos\(N_{1}+2 N_{2}=N\), y el factoring es único, hasta ordenar y reemplazar cualquiera\(b_{j} b y-b_{j}\).