Question

因式分解 x^3+8y^3+125z^3-30xyz

Answer 1

a^3+b^3+c^3-3abc

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

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

=(a+b)(a^2-ab+b^2)+c[(c^2-a^2-2ab-b^2)+(a^2-ab+b^2)

=(a+b)(a^2-ab+b^2)+c[c^2-(a+b)^2]+c(a^2-ab+b^2)

=(a+b+c)(a^2-ab+b^2)+c(a+b+c)(c-a-b)

=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)

x^3+8y^3+125z^3-30xyz

= x^3+(2y)^3+(5z)^3-30xyz

=(x+2y+5z)(x^2+4y^2+25z^2-2xy-10yz-5xz)