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Multiple root polynomial

Let \(c\) be a fixed real number.Show that a root of the equation \[x(x+1)(x+2).....(x+2014) = c\] can have multiplicity at most 2.Also determine the number of values of \(c\) for which the equation has a root of multiplicity \(2\).

Note by Eddie The Head
2 years, 8 months ago

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We can clearly assume \(c\neq 0\) - otherwise, there is nothing to prove, since all the roots are simple. If we suppose that \(x\) is a root of \(f(x)=x(x+1)\cdot\ldots\cdot(x+2014)-c\) with multiplicity three or more, we have that \(f(x)=0,f'(x)=0\) and \(f''(x)=0\). Since: \[ f'(x) = \left(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+2014}\right)\cdot x(x+1)\cdot\ldots(x+2014),\] \[ f''(x) = -\left(\frac{1}{x^2}+\frac{1}{(x+1)^2}+\ldots+\frac{1}{(x+2014)^2}\right)\cdot x(x+1)\cdot\ldots(x+2014)+\left(\frac{1}{x}+\frac{1}{x+1}+\ldots+\frac{1}{x+2014}\right)^2\cdot x(x+1)\cdot\ldots(x+2014),\] the three conditions give: \[ \sum_{j=0}^{2014}\frac{1}{(x+j)^2} = 0,\] that is impossible since all the terms are positive numbers.

The possible values of \(c\) for which a double root exists are given by \(y(y+1)\cdot\ldots\cdot(y+2014)\) where \(y\) is a root of \(\sum_{j=0}^{2014}\frac{1}{y+j}=0.\) Jack D'Aurizio · 2 years, 8 months ago

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Let the left side be \( f(x) \). First note that \( c = 0 \) doesn't work, as all the roots have multiplicity 1. Next, look at \( \frac{f'(x)}{f(x)} = \frac1{x} + \frac1{x+1} + \cdots + \frac1{x+2014} \). The right side is a decreasing function with \( 2015 \) vertical asymptotes; considering the graph, you can see that it crosses the \( x \)-axis once in every interval \( (-N-1,-N) \), where \( N \) is a nonnegative integer \( \le 2013 \). Since \( f'(x) \) is a polynomial of degree \( 2014 \), this shows that it has \( 2014 \) distinct roots.

Since \( f'(x) \) has no roots of multiplicity \( \ge 2 \), \( f(x) \) cannot have any roots of multiplicity \( \ge 3 \). So we're done with the first half.

This also shows that there are \( 2014 \) values \( a \) such that \( f(x) = f(a) \) has a root \( a \) of multiplicity \( 2 \). This might lead to fewer than \( 2014 \) values of \( c \), though. I don't think it does (but only because \( 2014 \) is even!). But my reasoning is probably better confined to a second post, mostly because I haven't fully worked it out yet. Patrick Corn · 2 years, 8 months ago

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It will take a lots of time to type the solution so I am writing the keypoints In order to have a multiplicity of 3 or more than that, the second derivative of the polynomial P(x) = x(x+1)....(x+2014) - c should be equal to zero. We can find the second derivative and apply Cauchy schwarz to know that it cannot be equal to zero. Siddharth Kumar · 2 years, 8 months ago

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