MHB What Is the Value of b If All Roots of the Polynomial Are Positive Real Numbers?

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The polynomial in question is \(x^4-8x^3+24x^2+bx+c=0\) and the goal is to determine the value of \(b\) under the condition that all roots are positive real numbers. The problem was presented as this week's Problem of the Week (POTW), following a previous week where no solutions were submitted. Community engagement is encouraged, with congratulations extended to Opalg for providing a correct solution. The discussion highlights the importance of finding conditions for polynomial roots, particularly focusing on positivity.
anemone
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Here is this week's POTW:

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If all the roots of $x^4-8x^3+24x^2+bx+c=0$ are positive reals, find the value of $b$.

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No one answered last week's POTW.(Sadface)

However, I have decided to give the community another week to attempt at the problem.(Smile)
 
Congratulations to Opalg for his correct solution (Cool) , which you can find below:

If $f(x) = x^4-8x^3+24x^2+bx+c$ has four real roots then its derivative must have three real roots. But $$f'(x) = 4x^3 - 24 x^2 + 48x + b = 4(x-2)^3 + b + 32,$$ and the function $(x-2)^3$ is strictly increasing except at the point $x=2$. So $f'(x)$ can only have three real roots if $b+32 = 0$. Therefore $b = -32$.
 
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