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Math Help - Roots Independent of lambda.

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    Roots Independent of lambda.

    If  x^5+5\lambda x^4+(\lambda a-4)x^2-(8\lambda+3)x+\lambda a-2=0. Then the value of a for which the Given equation has one root Independent of \lambda

    Here answer are
    (i) \frac{64}{5}

    (ii) -\frac{64}{5}

    (iii) 3

    (iv) -3
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  2. #2
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    Pick a value for lambda (1,2,3,...). Since this is multiple choice, you should be able to eliminate the wrong choices (plot the graphs).
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by jacks View Post
     x^5+5\lambda x^4+(\lambda a-4)x^2-(8\lambda+3)x+\lambda a-2=0. Then the value of a for which the Given equation has one root Independent of \lambda
    In general, for a family of polynomials:

    \{f_{\lambda,\;a}(x)=p(x)+\lambda q_a(x)\}

    if r is a root of p(x) and q_a(x) then, r is a root of f_{\lambda,\;a}(x). So, finding a root r of p(x) and substituting: q_a(r)=0 , we obtain a sufficient condition for a.

    That is a pity, if r=2 were a root of p(x) (it isn't) then,

    q_a(2)=0 \Leftrightarrow a=-64/5

    Possiibly (of course I'm not sure) there is typo.

    Fernando Revilla
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