在幫我解解一些有關因式分解的問題

Question

因式分解

(1)x^6-y^6
(2)(1-4a^2)(1-9b^2)+24ab
(3)(x^2-x+1)^2 -3(x^2-x)-7

Answer 1

(1)x^6-y^6
=(x^3)^2-(y^3)^2
=(x^3+y^3)(x^3-y^3)
=(x+y)(x^2-xy+y^2)(x-y)(x^2+xy+y^2)

(2)(1-4a^2)(1-9b^2)+24ab
=1-9b^2-4a^2+36a^2 b^2+24ab
=(1+12ab+36a^2 b^2)-(9b^2+4a^2-12ab)
=(6ab+1)^2-(2a-3b)^2
=(6ab+2a-3b+1)(6ab-2a+3b+1)

(3)(x^2-x+1)^2 -3(x^2-x)-7
=(x^2-x+1)^2 - 3(x^2-x+1)-4
=(x^2-x+1-4)(x^2-x+1+1)
=(x^2-x-3)(x^2-x+2)

Answer 2

(1)x6-y6=(x3)2-(y3)2=(x3-y3)(x3+y3)=(x-y)(x+y)(x2+xy+y2)(x2-xy+y2)(2)(1-4a2)(1-9b2)+24ab=1-4a2+24ab-9b2=1-(2a-3b)2=(1-2a+3b)(1+2a-3b)(3)(x2-x+1)2 -3(x2-x)-7 =(x2-x+1)2 -3(x2-x+1)-10=(x2-x+6)(x2-x-1)