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[Core] Polynomials

Polynomials

求解
New progress in senior mathematics compulsory part book 5 part 1 Ch10. 4  18 LetP(x)=x^(6)+ax^(5)+bx^(4)+cx^(3)+dx^(2)+ex+f.
If f is a prime number, how many distinct linear factors with integral
coefficients can P(x) at most have?



   

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Suppose P(x) can be rewritten as
(x+A)(x+B)(x+C)(x+D)(x+E)(x+F)
f = ABCDEF
Since f is prime,
five of the unknowns must be 1 / -1
and one of the unknowns must be f
P(x) = (x+1)^m (x-1)^n (x+f)
where m+n=5 and m, n are non negative integers.
Hence, if m=1, n=4, the number of linear factor of P(x) is the most.
P(x) = (x+1)(x-1)^4 (x+f) [3 linear factors]



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[ 本帖最後由 vikoau 於 2019-7-25 05:10 PM 編輯 ]

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