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Mathematik-Online problems:

Problem 107: Roots of a Polynomial

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Given the polynomial $ p(x)=2x^5-10x^4+10x^3+10x^2-12x-1$.
a)
Show, without using differential calculus, that all real roots of $ p$ are within the interval $ [-7,7]$. Divide therefor the equation $ p(x)=0$ by $ 2x^4$ and analyse the cases $ \vert x\vert\leq 5$ and $ \vert x\vert>5$.
b)
Find the values of the function $ p$ for $ x_k=\frac{k}{2},\, k\in\{-2,\,-1,\,\ldots , 7\}$. In which subintervals $ I_k=[x_k, x_{k+1}]$ are the roots - and why?
c)
Find, by successive bisection of the starting intervals $ I_k$ determined in b), the intervals of length $ \frac{1}{8}$ that contain each exactly one root of $ p$.

(Authors: Apprich/Höfert)

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[Solutions]
  automatisch erstellt am 14. 10. 2004