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Algebra

Scorching goIITian

 Joined: 18 Feb 2012 Post: 264
2 May 2012 22:53:44 IST
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interesting@!!
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

suppose a1,a2,a3...(n-1)trms are non negative real nos. and cosider the poly.p(x)=x^n+a1x^n-1+a2x^n-2......+1.assume p(x) has n real roots.prove that p(2)>=3^n??

Forum Expert
Joined: 18 Dec 2007
Posts: 934
3 May 2012 21:46:29 IST
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We can write,

p(x)= (x+b1)(x+b2)... (x+bn) where all bi>0 and b1b2...bn=1

Now, apply AM-GM inequality and theory of equations to conclude that

ai>=nCi for i=1,2,...,n

Thus p(2)>= 2^n+nC1 2^(n-1)+ ... +1= 3^n

Scorching goIITian

Joined: 18 Feb 2012
Posts: 264
4 May 2012 15:14:23 IST
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YES ABHISEKH SIR I HAV DONE IN THE SAME WAY.

Blazing goIITian

Joined: 19 Jan 2008
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4 May 2012 18:42:55 IST
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Me also done in same wayWhen I saw that Abhishek Sir has posted it,. then I have leave my post

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4 May 2012 23:03:47 IST
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Blazing goIITian

Joined: 19 Jan 2008
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4 May 2012 23:07:59 IST
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Great SirNice solution

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4 May 2012 23:10:23 IST
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Forum Expert
Joined: 18 Dec 2007
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5 May 2012 00:11:07 IST
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