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(x^2-y^2)^3+(y^2-z^2)^3+(z^2-x^2)^3怎么分解因式
题目详情
(x^2-y^2)^3+(y^2-z^2)^3+(z^2-x^2)^3怎么分解因式
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答案和解析
a=x²-y²,b=z²-x²
原式=a³+(-a-b)³+b³
=(a+b)(a²-ab+b²)-(a+b)³
=(a+b)(a²-ab+b²-a²-2ab-b²)
=-3ab(a+b)
=-3(x²-y²)(z²-x²)(x²-y²+z²-x²)
=-3(x+y)(x-y)(z+x)(z-x)(z+y)(z-y)
原式=a³+(-a-b)³+b³
=(a+b)(a²-ab+b²)-(a+b)³
=(a+b)(a²-ab+b²-a²-2ab-b²)
=-3ab(a+b)
=-3(x²-y²)(z²-x²)(x²-y²+z²-x²)
=-3(x+y)(x-y)(z+x)(z-x)(z+y)(z-y)
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