初中代数题求解

[huangxinru]

已知a、b为正数，且a[a(a^3+a^2*b+a*b+b)+b]+b=1,求a+b的值。

[ufouser]

由上式得
a^5 + a^4 * b + a^3 * b + a^2 * b + a * b + b = 1
==>
(a^5 - 1) + b(a^4 + a^3 + a^2 + a^1 + a^0) = 0
==>
(1).若a=1代入得b=0 ==> a+b等于1
(2).若a不等于1, 则a-1不等于0
==>(a^5 - 1) + b(a^4 + a^3 + a^2 + a^1 + a^0)) * (a-1) / (a-1) = 0
==>(a^5 - 1) + b(a^5 - 1) / (a - 1) = 0
==>(a^5 - 1)[1 + b/(a-1)] = 0
==>(a^5 - 1)[(a-1+b)/(a-1)] = 0
==>(a^5 - 1)(a-1+b) = 0 (因为a-1不等于0 ==>a^5-1不等于0)
==>a - 1 + b == 0
==> a+b == 1

[huangxinru]

强，100分归你