اتحاد شرطی 1
اگر a + + c = 0 باشد، ثابت کنید:
(a2 + b2 + c2)2 = (a4 + b4 + c4)
راه اول
a + b + c = 0 => 0 = (a + b + c)2 = a2 + b2 + c2 + 2ab + bc + 2ac
=> a2 + b2 + c2 = -2ab -2bc - ca
=> (a2 + b2 + c2)2 = (2ab + 2bc + 2ca)2=
= 4a2b2 + 4b2c2 + 4c2a2 + 8ab2c + 8abc2 + 8a2bc =
4a2b2 + b2c2 + 4c2a2 + 8abc (a + b + c) =>
(a2 + b2 + c2)2 = 4a2 b2 + 4b2c2 + 4c2a2
(a2 + b2 + c2)2 = a4 + b4 + c4 + 2a2b2 + 2b2c2 + 2c2a2
(a2 + b2 + c2)2 = a4 + b4 + c4 + (a2 + b2 + c2)2 =>
(a2 + b2 + c2)2 = (a4 + b4 + c4)
راه دوم
a + + c = 0 => c = -a -b =>
(a2 + b2 + c2)2 = (a2 + b2 + (- -b)2)2 = (2a2 + 2b2 + 2ab)2 =
= 4a4 + 4b4 + 4a2b2 + a2b2 + 8a3b + 8ab3 =
= 2a4 + 2b4 + 2(a4 + b4 + 4a3b + a2b2 + 4ab3) =
= 2a4 + 2b4 + 2(-a -b)4 = 2a4 + 2b4 + 2c4
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