因式分解(高手請進)--20點!!

1) 3(3x^3 + 10y^3) +10z(10z^2 - 9xy)
= 9x^3 + 30y^3 + 100z^3 - 90xyz
= 9x^3 + 30y^3 + 100z^3 - 3 ∛[(9x^3)(30y^3)(100z^3)]
令 a^3 = 9x^3 , b^3 = 30y^3 , c^3 = 100z^3 ,
原式
= a^3 + b^3 + c^3 - 3abc

= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
= [(∛9)x + (∛30) y + (∛100)z] [(3∛3)x^2 + (∛900)y^2 + (10∛10)z^2
- (3∛10)xy - (10∛3)yz - (∛900)zx]

2)43719x^2 - 87276x + 19672

= 43719{ x - [87276 + √(87276^2 - 4*43719*19672)]/(2 * 43719) } *
{ x - [87276 - √(87276^2 - 4*43719*19672)]/(2 * 43719) }

= 43719(x - 1.737290955...)(x - 0.259003562...)

3) x^4 - 8x^2 +16
= (x^2 - 4)^2
= [(x - 2)(x + 2)]^2

4)x^8 + (x^4)(y^4) + y^8
= x^8 + 2(x^4)(y^4) + y^8 - (x^4)(y^4)
= (x^4 + y^4)^2 - [(x^2)(y^2)]^2
= [x^4 + y^4 - (x^2)(y^2)] [x^4 + y^4 + (x^2)(y^2)]
= [x^4 + y^4 - (x^2)(y^2)] [x^4 + y^4 + 2(x^2)(y^2) - (x^2)(y^2)]
= [x^4 + y^4 - (x^2)(y^2)] [(x^2 + y^2)^2 - (xy)^2]
= (x^2 + y^2 - xy)(x^2 + y^2 + xy) [x^4 + y^4 - (x^2)(y^2)]

5)x^4 -4(a^2)(x^2)- 12abx - 9b^2
= x^4 - (2ax + 3b)^2
= (x^2 - 2ax - 3b)(x^2 + 2ax + 3b)

6) 2x^2 - xy - y^2 - 7x + y +6
= (2x + y)(x - y) - [2(2x + y) + 3(x - y)] + (-2)(-3)
= (2x + y - 3)(x - y - 2)

7) (x-1)(x-3)(x-5)(x-7)-9
= (x-1)(x-7)(x-3)(x-5) - 9
= (x^2 - 8x + 7)(x^2 - 8x + 15) - 9
令 x^2 - 8x = k ,
= (k + 7)(k + 15) - 9
= k^2 + 22k + 96
= (k + 6)(k + 16)
= (x^2 - 8x + 6)(x^2 - 8x + 16)
= (x^2 - 8x + 6)(x - 4)^2

8) (y^2)(z^2)(x^4 -1) + (x^2)(y^4 -z^4)
= (y^2)(z^2) x^4 + (y^4 - z^4)x^2 - (y^2)(z^2)
= [(y^2)(x^2) - z^2] [(z^2)(x^2) + y^2)
= (yx - z) (yx + z) [(z^2)(x^2) + y^2)

9) 9x^4 +15x^3 -35x^2 + 5x + 6
因 9 + 15 - 35 + 5 + 6 = 0
(x - 1) 是因式之一 ,
= (x - 1)(9x^3 + 24x^2 - 11x - 6)
= (x - 1)(x + 3)(9x^2 - 3x - 2)
= (x - 1)(x + 3)(3x + 1)(3x - 2)

10) 3a^2 - 11ab + 6b^2 - ac - 4bc - 2c^2
= (3a - 2b)(a - 3b) - ac - 4bc - 2c^2
設原式 = (3a - 2b + Mc)(a - 3b + Nc)
= (3a - 2b)(a - 3b) + Nc(3a - 2b) + Mc(a - 3b) + MNc^2
= (3a - 2b)(a - 3b) + (3N+M)ac - (2N+3M)bc + MNc^2
比較系數 :
3N+M = - 1
2N+3M = 4
M = 2
N = -1
原式 = (3a - 2b + Mc)(a - 3b + Nc)
= (3a - 2b + 2c)(a - 3b - c)

3) x^4 - 8x^2 +16
= (X^2 - 4)^2
= (X-2)^2 (X+2)^2

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2010-06-11 17:45:42 補充：
4) x^8 + (x^4)(y^4) + y^8
= (x^4+y^4)^2 - (x^4)(y^4)
= (x^4 + y^4 – x^2 y^2) (x^4 + y^4 + x^2 y^2)


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