因式分解3題

(1) 4a^2 - b^2 + c^2 - 9d^2 + 4ac +6bd= (4a^2 + 4ac + c^2) - (b^2 - 6bd + 9d^2)= (2a + c)^2 - (b - 3d)^2= (2a - b + c + 3d)(2a + b + c - 3d)
2) (a+b)^3 + (b+c)^3 + (c+a)^3 + a^3 + b^3 + c^3= [(a+b)^3 + c^3] + [(b+c)^3 + a^3] + [(c+a)^3 + b^3]= (a+b+c)[(a+b)^2 - (a+b)c + c^2] + (a+b+c)[(b+c)^2 - (b+c)a + a^2]
+ (a+b+c)[(c+a)^2 - (c+a)b + b^2]= (a+b+c)[(a+b)^2 + (b+c)^2 + (c+a)^2 - (a+b)c - (b+c)a - (c+a)b + a^2 + b^2 + c^2]= (a+b+c)[(a+b)^2 + (b+c)^2 + (c+a)^2 - 2ab - 2bc - 2ca + a^2 + b^2 + c^2]= (a+b+c)[a^2 + b^2 + b^2 + c^2 + c^2 + a^2 + a^2 + b^2 + c^2]= 3(a + b + c)(a^2 + b^2 + c^2)
3) (a+b)^2 + (a+c)^2 - (c+d)^2 - (b+d)^2 = (a+b)^2 - (c+d)^2 + (a+c)^2 - (b+d)^2= (a+b+c+d)(a+b-c-d) + (a+b+c+d)(a+c-b-d)= (a+b+c+d)(a+b-c-d + a+c-b-d)= (a+b+c+d)(2a-2d)= 2(a - d)(a + b + c + d)

(4a^2+4ac+c^2)-(b^2-6bd+9d^2)
(2a+c)^2-(b-3d)^2
(2a+c-b+3d)(2a+c+b-3d)

2010-08-12 22:23:16 補充：
(a+b)^2 + (a+c)^2 - (c+d)^2 - (b+d)^2
(a+b-c-d)(a+b+c+d)-(b+d-a-c)(b+d+a+c)
(a+b+c+d)(a+b-c-d-b-d+a+c)
(a+b+c+d)(2a-2d)
2(a-d)(a+b+c+d)

(1) 4a^2 - b^2 + c^2 - 9d^2 + 4ac +6bd
= (4a^2 + 4ac + c^2) - (b^2 - 6bd + 9d^2)
= (2a + c)^2 - (b - 3d)^2
= (2a-b+c+3d)(2a+b+c-3d)

2010-08-12 21:44:57 補充：
(3) (a+b)^2 + (a+c)^2 - (c+d)^2 - (b+d)^2
= (a+b-c-d)(a+b+c+d) + (a+b+c+d)(a-b+c-d)
= (a+b+c+d)(a+b-c-d+a-b+c-d)
= (a+b+c+d)(2a-2d)
= 2(a-d)(a+b+c+d)