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If $$\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+\ldots+\frac{a_{n-1}}{2}+a_{n}=0,$$ then the maximum possible number of roots of the equation $${a_0}{x^n}+{a_1}{x^{n-1}}+{a_2}{x^{n-2}}+\ldots+{a_{n-1}}{x}+{a_n}=0$$ in $(0,1)$ will be...?

This seems like a really interesting question. Not exactly within my XII class syllabus, but I want to solve it anyway. Any hints on how I could go about it.

Andreas Caranti
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sammy2014
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1 Answers1

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The answer is $n$.

Indeed, let $P_n(X)=\sum_{k=0}^na_{n-k}x^{k}$ be the Legendre polynomial of degree $n$. It is proportional to $\dfrac{d^n}{dx^n}(x^n(1-x)^n)$. It is well-known that $P_n$ has $n$ distinct zeros in the interval $(0,1)$, and by orthogonality we have also $$\sum_{k=0}^n\frac{a_{n-k}}{k+1}=\int_0^{1}P_n(t)dt=0.$$

Omran Kouba
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