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Polynomial Elegance (Posted on 2024-08-19)

Let's say α = (11/5)^1/7 + (5/11)^1/7 is the root of a polynomial P(x) where the coefficient of the highest degree of the polynomial is -55 and all the coefficients are integers. Calculate P(1).

No Solution Yet Submitted by Danish Ahmed Khan

Rating: 5.0000 (1 votes)

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Solution

Comment 1 of 1

For simplicity a=(11/5)^(1/7) and b=(5/11)^(1/7)

x= α = a+b

Each of the following simplifies nicely since ab=1

x^3= a^3 + 3a + 3b + b^3

x^5= a^5 + 5a^3 + 10a + 10b + 5b^3 + b^5

x^7= a^7 + 7a^5 + 21a^3 + 35a + 35b + 21b^3 + 7b^5 + b^7

x^7 - 7x^5 + 14x^3 - 7x = a^7 + b^7 = 11/5 + 5/11 = 146/55

P(x) = -55x^7 + 385x^5 - 770x^3 +385x + 146

P(1) = -55+385-770+385+146 = 91

This reminds me of Two Mini-Pollies from a few days ago but required more creativity

http://perplexus.info/show.php?pid=13937

Posted by Jer on 2024-08-19 14:07:54

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