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[SPAM] [obm-l] Polinômios de variável complexa
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Esta quest=E3o foi da prova de =E1lgebra do IME=20
1976/1977. Vou transliterar um pouco o enunciado.
Seja P(x)=3D(x+1)(x+3)(x+5)+k(x+2)(x+4), com x=20
complexo e k real positivo. Desenhar no plano=20
complexo o lugar geom=E9trico das ra=EDzes de P(x)=3D0=20
para todos os valores poss=EDveis de k.
Tentei o seguinte: se z=3Da+bi =E9 raiz de P(x),=20
ent=E3o P(z)=3D0, o que implica que Re[P(z)]=3D0 e=20
Im[P(z)]=3D0, ent=E3o daria para obter express=F5es em=20
fun=E7=E3o de a e b que descrevessem o lugar=20
geom=E9trico procurado. S=F3 que as express=F5es parecem intrat=E1veis.
Alguma outra id=E9ia?
[ ]'s
J. R. Smolka =20
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<body>
Esta quest=E3o foi da prova de =E1lgebra do IME 1976/1977. Vou transliterar
um pouco o enunciado.<br><br>
Seja P(x)=3D(x+1)(x+3)(x+5)+k(x+2)(x+4), com x complexo e k real positivo.
Desenhar no plano complexo o lugar geom=E9trico das ra=EDzes de P(x)=
=3D0
para todos os valores poss=EDveis de k.<br><br>
Tentei o seguinte: se z=3Da+bi =E9 raiz de P(x), ent=E3o P(z)=3D0, o que imp=
lica
que Re[P(z)]=3D0 e Im[P(z)]=3D0, ent=E3o daria para obter express=F5es em fu=
n=E7=E3o
de a e b que descrevessem o lugar geom=E9trico procurado. S=F3 que as
express=F5es parecem intrat=E1veis.<br><br>
Alguma outra id=E9ia?<br><br>
[ ]'s<br><br>
<x-sigsep><p></x-sigsep>
<font size=3D4><b>J. R. Smolka</b></font> </body>
</html>
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