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Septic Factorization (Posted on 2015-06-07) Difficulty: 3 of 5
Factorize: (a+b)7 - a7 - b7

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No Solution Yet Submitted by K Sengupta    
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another approach Comment 5 of 5 |
let f(a,b)=(a+b)^7-a^7-b^7

now f(0,b)=0 and f(a,0)=0 thus we have
where k is a constant and g(a,b) is a polynomial in a,b

now if a=-b then f(a,b)=0 as well thus we can simplify further to

again with k a constant and g(a,b) a polynomial in a,b
now by expanding f(a,b) and dividing out ab(a+b) we can see that
k=7 and g(a,b)=a^4+2a^3b+3a^2b^2+2ab^3+b^4
now this is where a bit of intuition comes into play.  The symmetry in the coefficients makes me think of squared polynomial thus we have
now we can deduce that h(a,b) is a polynomial in a,b of degree two with 3 terms two of which are a^2 and b^2 and since the remaining terms of g(a,b) have a common factor of ab then the third term of h(a,b) must be ab
thus we can check if
which by expanding is verified

thus we are left with

  Posted by Daniel on 2015-06-08 11:47:38
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