Given that $f(x) :=x^6-6x^5+ax^4+bx^3+cx^2+dx+1$ has its roots as all positive, find $a,b,c,d$

Asked Jul 20 '18 at 18:42

Active Jul 20 '18 at 18:42

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From Larson's "Problem Solving Through Problems" 7.2.10:

Given that $f(x) :=x^6-6x^5+ax^4+bx^3+cx^2+dx+1$ has its roots as all positive, find $a,b,c,d$

Thus chapter is about (generalized) Arithmetic-geometric mean inequality so I would have to use that. I believe it's implied from the problem that all roots are real.

Any hints or solution would be great.

asked Jul 20 '18 at 18:42

Squirrel-Power

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$6$ is the sum of the roots and $1$ the product. Look at the condition when the arithmetic mean is equal to the geometric mean. Not just knowing that the roots are real, but positive. – Jul 20 '18 at 18:44
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I got it, since the sum of roots divided by $6$ is actually equal to the geometric mean of the roots, it must be that they are all equal to $1$ and hence we just have to expand $(x-1)^6$ – Squirrel-Power Jul 20 '18 at 18:49

Given that $f(x) :=x^6-6x^5+ax^4+bx^3+cx^2+dx+1$ has its roots as all positive, find $a,b,c,d$

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