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manpreet Best Answer 3 years ago

a 0 n + 1 + a 1 n + a 2 n - 1 + \dots + 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 + \dots + a n - 1 x + a 111 " class="mi" style="margin: 0px; padding: 0px;

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manpreet 3 years ago

The answer is $n$ .

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

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Quadratic expression

Manpreet Singh

Answers (2)

manpreet Best Answer 3 years ago

manpreet 3 years ago

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