Question

因式分解：（x^5+x^4+x^3+x^2+x+1)^2-x^5

Answer 1

(x^5+x^4+x^3+x^2+x+1)^2-x^5
=(x^6-1)^2/(x-1)^2-x^5
=[x^12-2x^6+1-x^7+2x^6-x^5]/(x-1)^2
=[(x^7-1)(x^5-1)]/(x-1)^2
=(x^6+x^5+x^4+x^3+x^2+x+1)(x^4+x^3+x^2+x+1)

追问

上面一步是怎么到下面一步的？

追答

=(x^6-1)^2/(x-1)^2-x^5

=(x^6-1)^2 - x^5 (x-1)^2 / (x-1)^2
分解开再合并同类项

=[x^12-2x^6+1-x^7+2x^6-x^5]/(x-1)^2

追问

配方时分组是怎么分的？

追答

=[x^12-2x^6+1-x^7+2x^6-x^5]/(x-1)^2

=[x^12 -x^7 -x^5 +1 ] /(x-1)^2

=[(x^7-1)(x^5-1)]/(x-1)^2

Answer 2

原式=[x^5+(x^4+x^3+x^2+x+1)]^2-x^5
=(x^5)^2+2(x^4+x^3+x^2+x+1)x^5+(x^4+x^3+x^2+x+1)^2-x^5
=(x^5)^2+(2x^4+2x^3+2x^2+2x+1)x^5+(x^4+x^3+x^2+x+1)^2
=(x^5+2x^4+2x^3+2x^2+2x+1)x^5+(x^4+x^3+x^2+x+1)^2
=[(x^5+x^4+x^3+x^2+x)+(x^4+x^3+x^2+x+1)]x^5+(x^4+x^3+x^2+x+1)^2
=[x(x^4+x^3+x^2+x+1)+(x^4+x^3+x^2+x+1)]x^5+(x^4+x^3+x^2+x+1)^2
=(x^4+x^3+x^2+x+1)(x+1)x^5+(x^4+x^3+x^2+x+1)^2
=(x^4+x^3+x^2+x+1)(x^6+x^5+x^4+x^3+x^2+x+1)