form2 數[因式分解] 急!!! 急!!! 急!!! 20點!!!

1)(x + y)2﹣(y + 1)2
= [(x + y)﹣(y + 1)][(x + y) + (y + 1)]
= (x + y﹣y﹣1)(x + y + y + 1)
= (x﹣1)(x + 2y + 1)

2)8m2﹣24mn + 18n2
= 2(4m2﹣12mn + 9n2)

= 2[(2m)2﹣2(2m)(3n) + (3n)2]
= 2(2m﹣3n)2

3)1﹣2a2 + a4
= (a2)2﹣2a2 + 1
= (a2﹣1)2

4)x2﹣14x﹣72，利用十字相乘法
x ↘ -18
x ↖ 4
===========
= (x﹣18)(x + 4)

5)4x2﹣4x﹣15，利用十字相乘法
2x ↘ -5
2x ↖ 3
==============
= (2x﹣5)(2x + 3)

6)3x2﹣7x﹣6
3x ↘ 2
x ↖ -3
==============
= (3x + 2)(x﹣3)

7)2x2﹣22x + 48
2x ↘ -16
x ↖ -3
==============
= (2x﹣16)(x﹣3)

8)6x3﹣48y3
= 6(x3﹣8y3)
= 6[x3﹣(2y)3]
= 6(x﹣2y)(x2 + 2xy + 4y2)
= 6(x﹣2y)(x + 2y)2

9)54y3+2x3 z6
= 2[(3y)3 + (xz2)3]
= 2(3x + xz2)(9y2﹣3xyz2 + x2 z4)



10a)x4﹣6x2 + 9

= (x2 )2 ﹣2(3)(x2) + 32
= (x2﹣3)2
10b)x4﹣6x2 + 9﹣4x2
= (x2﹣3)2﹣4x2
= (x2﹣3﹣4x)(x2﹣3 + 4x)
= (x2﹣4x﹣3)(x2 + 4x﹣3)


11)x6﹣1

= (x3 + 1)(x3﹣1)

= (x + 1)(x2﹣x + 1)(x﹣1)(x2 + x + 1)

= (x + 1)(x﹣1)(x2 + x + 1)(x2﹣x + 1)

12a)x4 + x2 + 1

= (x4 + 2x2 + 1)﹣x2

= (x2 + 1)2﹣x2

= (x2 + 1﹣x)(x2 + 1 + x)

= (x2 + x + 1)(x2﹣x+1)

12b)利用12a的結果,
(x + 1)4+ (x + 1)2 + 1

= [(x + 1)2 + (x + 1) + 1][(x + 1)^2﹣(x + 1) + 1]

= (x2 + 2x + 1 + x + 1 + 1)(x2 + 2x + 1﹣x﹣1 + 1)

= (x2 + 3x + 3)(x2 + x + 1)

2008-08-13 19:05:18 補充：
利用恆等式：
a^2 - b^2 = (a - b)(a + b)
(a - b)^2 = a^2 - 2ab + b^2
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
a^3 + b^3 = (a+b)(a^2 - ab + b^2)