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Find the number of real roots of \[2x^{2001} + 3x^{2000} + 2x^{1999} + 3x^{1998} + \dots + 2x + 3 = 0.\]

 Apr 12, 2020
 #1
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There are 5 real roots.

 Apr 13, 2020
 #2
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\(2x^{2001} + 3x^{2000} + 2x^{1999} + 3x^{1998} + \cdots + 2x + 3 = 0\\ (2x + 3)(x^{2020} + x^{2018} + \cdots + 1) = 0\)

Because \(x^{2020} + x^{2018} + \cdots + 1 > 0\), the only possibility is \(2x + 3 = 0\).

Solving gives \(x = -\dfrac32\).
There is only 1 real root.

 Apr 14, 2020

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