Question

Ajuda ae ????????????????

Answer 1

$x^{3}-6x^{2}-kx-30=0\\\\\boxed{a=1~~~~~~b=-6~~~~~~c=-k~~~~~~d=-30}$

Sejam as 3 raízes da equação α₁, α₂ e α₃

Consideremos que α₁ + α₂ = 1 (dado do exercício)
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Achando a soma das raízes:

$S=-b/a\\\alpha_{1}+\alpha_{2}+\alpha_{3}=-(-6)/1\\\alpha_{1}+\alpha_{2}+\alpha_{3}=6\\1+\alpha_{3}=6\\\alpha_{3}=5$

Achando o produto das raízes:

$P=-d/a\\\alpha_{1}\cdot\alpha_{2}\cdot\alpha_{3}=-(-30)/1\\\alpha_{1}\cdot\alpha_{2}\cdot5=30\\\alpha_{1}\cdot\alpha_{2}=6$

Outra relação de girard:

$\alpha_{1}\cdot\alpha_{2}+\alpha_{1}\cdot\alpha_{3}+\alpha_{2}\cdot\alpha_{3}=c/a\\\alpha_{1}\cdot\alpha_{2}+\alpha_{1}\cdot\alpha_{3}+\alpha_{2}\cdot\alpha_{3}=-k/1\\\alpha_{1}\cdot\alpha_{2}+\alpha_{1}\cdot\alpha_{3}+\alpha_{2}\cdot\alpha_{3}=-k\\6+\alpha_{1}\cdot\alpha_{3}+\alpha_{2}\cdot\alpha_{3}=-k\\6+\alpha_{3}(\alpha_{1}+\alpha_{2})=-k\\6+5(1)=-k\\6+5=-k\\-k=11\\\\\boxed{\boxed{k=-11}}$