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Let  r1, r2, r3 and r4 be the roots of x^4 - 2x^3 - 5x^2 + 4x - 1 = 0
Find the monic polynomial , in x,  whose roots are 1/r1, 1/r2, 1/r3 and 1/r4 

 Mar 15, 2020
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Let  r1, r2, r3 and r4 be the roots of x^4 - 2x^3 - 5x^2 + 4x - 1 = 0
Find the monic polynomial , in x,  whose roots are 1/r1, 1/r2, 1/r3 and 1/r4 

 

\(\begin{array}{|rcll|} \hline \mathbf{x^4 - 2x^3 - 5x^2 + 4x - 1} &=& \mathbf{0} \\\\ r^4 - 2r^3 - 5r^2 + 4r - 1 &=& 0 \quad & | \quad : r^4 \\ 1 - 2\dfrac{1}{r} - 5\dfrac{1}{r^2} + 4\dfrac{1}{r^3} - \dfrac{1}{r^4} &=& 0 \quad & | \quad x= \dfrac{1}{r} \\ 1 - 2x - 5x^2 + 4x^3 - x^4 &=& 0 \quad & | \quad *(-1) \\ \mathbf{x^4 -4x^3 + 5x^2+2x-1} &=& \mathbf{0} \\ \hline \end{array}\)

 

\(\mathbf{x^4 - 2x^3 - 5x^2 + 4x - 1=0} \\ \text{Real solutions}: \\ x=-1.89614823140755 \\ x=3.20188000352314 \\ \text{Complex solutions}: \\ x=0.347134113942206 - 0.210259195466214 i \\ x=0.347134113942206 + 0.210259195466214 i\)

 

\(\mathbf{x^4 - 4x^3 + 5x^2 + 2x - 1 = 0} \\ \text{Real solutions}: \\ x=-0.527384928792028 \\ x=0.312316513704344 \\ \text{Complex solutions}: \\ x=2.10753420754384 + 1.27653384988106 i \\ x=2.10753420754384 - 1.27653384988106 i\)

 

laugh

 Mar 16, 2020
edited by heureka  Mar 16, 2020

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