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Rugownik - Wikipedia, wolna encyklopedia

Rugownik

Z Wikipedii

W matematyce rugownikiem dwóch wielomianów

P(x)=a_{n} x^{n}+a_{1} + a_{n-1} x^{n-1} + a_{2} x^{2} + \dots + a_{0}
Q(x)=b_{m} x^{m}+b_{1} + b_{m-1} x^{m-1} + b_{m-2} x^{m-2} + \dots + b_{0}

nazywa się wyznacznik stopnia n + m postaci

\mathrm{R}(P,Q)=\begin{vmatrix} a_{n} & a_{n-1} & a_{n-2} & \dots & a_{0} & 0 & 0 & \dots & 0 \\ 0 & a_{n} & a_{n-1} & \dots & a_{1} & a_{0} & 0 & \dots & 0 \\ 0 & 0 & a_{n} & \dots & a_{2} & a_{1} & a_{0} & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & a_{n} & a_{n-1} & a_{n-2} & \dots & a_{0} \\ b_{m} & b_{m-1} & b_{m-2} & \dots & b_{0} & 0 & 0 & \dots & 0 \\ 0 & b_{m} & b_{m-1} & \dots & b_{1} & b_{0} & 0 & \dots & 0 \\ 0 & 0 & b_{m} & \dots & b_{2} & b_{1} & b_{0} & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & b_{m} & b_{m-1} & b_{m-2} & \dots & b_{0} \end{vmatrix}.

Dla dowolnych wielomianów P,Q,R:

  • \mathrm{R}(P,Q) = (-1)^{\deg P \cdot \deg Q} \cdot \mathrm{R}(Q,P)
  • \mathrm{R}(P\cdot R,Q) = \mathrm{R}(P,Q) \cdot \mathrm{R}(R,Q)
  • R(P,Q) = 0 wtedy i tylko wtedy, gdy P\, i Q\, mają wspólny pierwiastek.

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