Question

分解因式：(b+c-2a)^3+(c+a-2b)^3+(a+b-2c)^3

Answer 1

设a-b=d,b-c=e,c-a=f

(b+c-2a)^3+(c+a-2b)^3+(a+b-2c)^3

=(f-d)^3+(d-e)^3+(e-f)^3

=(f-e)((f-d)^2-(f-d)(d-e)+(d-e)^2)-(f-e)^3

=(f-e)((f-d)^2-(f-d)(d-e)+(d-e)^2-(f-e)^2)

=(f-e)((f-d)(f-d-d+e)+(d-e-f+e)(d-e+f-e))

=(f-e)((f-d)(f+e-2d)-(f-d)(f+d-2e))

=(f-e)(f-d)(f+e-2d-f-d+2e)

=(f-e)(f-d)(3e-3d)

=3(f-e)(f-d)(e-d)

=3(c-a-(b-c))(c-a-(a-b))(b-c-(a-b))

=3(2c-a-b)(b+c-2a)(2b-c-a)