Find the factors of (a+b+c)4−(b+c)4−(c−a)4−(a+b)4+a4+b4+c4.
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Solution
Given expression is (a+b+c)4−(b+c)4−(c+a)4−(a+b)4+a4+b4+c4
Put b=0 in the given expression, (a+0+c)4−(0+c)4−(c+a)4−(a+0)4+a4+04+c4=0 Similarly, we get expression value as '0' for a=0 and c=0. Put a+b+c=0,c+a=−b,b+a=−c,b+c=−a in the given expression (0)4−(−a)4−(−b)4−(−c)4+a4+b4+c4=0 As it is fourth power equation, we can it write as (a+b+c)4−(b+c)4−(c+a)4−(a+b)4+a4+b4+c4=kabc(a+b+c) Put a=1,b=2,c=3, we get (1+2+3)4−(2+3)4−(3+1)4−(1+2)4+14+24+34=k×1×2×3×(1+2+3) On simplification, we get 432=k(36) k=12 Therefore, factors are 12abc(a+b+c)