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determine all positive integers n such that the polynomial with n+1 terms f(x)=(xpower 4n)+xpower 4(n-1)+....+xpower8+(xpower4)+1 is divisible by g(x)=(xpower 2n)+xpower 2(n-1)+....+xpower4+xpower2)+1.

determine all positive integers n such that the polynomial with n+1 terms f(x)=(xpower 4n)+xpower 4(n-1)+....+xpower8+(xpower4)+1 is divisible by g(x)=(xpower 2n)+xpower 2(n-1)+....+xpower4+xpower2)+1.

Grade:10

2 Answers

Chetan Mandayam Nayakar
312 Points
11 years ago

both f(x) and g(x) are geometric progressions. for f(x) and g(x),a=1,for f(x), r=x4, for g(x), it is x2

f(x)/g(x)=((1-x4(n+1))/(1-x4))((1-x2(n+1))/(1-x2))=(1+x2(n+1))/(1+x2), it is clear from algebra that n+1 is odd, implying that n is any even natural number

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Chetan Mandayam Nayakar
312 Points
11 years ago

both the num(numerator) and den(denominator) are simple geometric progressions

num/den= ((1-x4(n+1))/(1-x4))(1-x2)/(1-x2(n+1)) = (1+x2(n+1))/(1+x2)

obviously n+1 is odd which implies that n is any even natural number

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