F.2 Maths Factorization (急)

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If you have any questions, please feel free to ask.

1)
6a^2 - 13ab + 6b^2 - 4a + b - 2
唔明? 有否抄錯?


2)
27(a+b)^2 - 12(a-b)^2
= 3 [ 9(a+b)^2 - 4(a-b)^2 ]
= 3 { [3(a+b)]^2 - [2(a-b)]^2 }
= 3 [3(a+b)+2(a-b)] [3(a+b)-2(a-b)]
= 3 (3a+3b+2a-2b) (3a+3b-2a+2b)
= 3 (5a+b) (a+5b)

3)
25(x^2+2)^2 - (5x^2+20x)^2
= 25(x^2+2)^2 - (5^2)[(x^2+4x)]^2
= 25 [ (x^2+2)^2 - (x^2+4x)^2 ]
= 25 [ (x^2+2)+(x^2+4x) ] [ (x^2+2)-(x^2+4x)]
= 25 (x^2+2+x^2+4x) (x^2+2-x^2-4x)
= 25 (2x^2+4x+2) (2-4x)
= 25*2*2*(x^2+2x+1)(1-2x)
= 100*(x+1)^2*(1-2x)

4)
a^4 + 4
= (a^2)^2 + 2^2
= (a^2)^2 + 2(a^2)(2) + 2^2 - 2(a^2)(2)
= (a^2+2)^2 - 4a^2
= (a^2+2)^2 - (2a)^2
= [(a^2+2)+2a] [(a^2+2)-2a]
= (a^2+2a+2) (a^2-2a+2)

5)
(1-a^2)(1-b^2) - 4ab
= 1-b^2-a^2+(a^2)(b^2) - 4ab
= (a^2)(b^2) - 2(ab)(1) + 1 - 2ab - b^2 - a^2
= (ab-1)^2 - (a^2+2ab+b^2)
= (ab-1)^2 - (a+b)^2
= [(ab-1)+(a+b)] [(ab-1)-(a+b)]
= (ab+a+b-1)(ab-a-b-1)

6)
(a-b)(a+b)^3 - a^4 + b^4
= (a-b)(a+b)(a+b)^2 - [(a^2)^2 - (b^2)^2]
= (a^2-b^2)(a+b)^2 - [(a^2+b^2)(a^2-b^2)]
= (a^2-b^2) [(a+b)^2 - (a^2+b^2)]
= (a^2-b^2) (a^2+2ab+b^2-a^2-b^2)
= (a^2-b^2) (2ab)
= 2ab(a+b)(a-b)

7)
9a^2 - b^2 + 3ab^2 - 1
= [(3a)^2 - 1] - (3ab^2 -b^2)
= (3a+1)(3a-1) - b^2(3a-1)
= (3a-1) (3a-1-b^2)

8)
x (a^3+b^3) + ba (a^2 - b^2) + b^3 (a+b)
= x(a^3) + x(b^3) + b(a^3) - a(b^3) + (a+b)(b^3)
= (x+b)(a^3) + (x-a+a+b)(b^3)
= (x+b)(a^3) + (x+b) (b^3)
= (x+b) (a^3+b^3)
= (x+b) {[a^3+3(a^2)(b)+3a(b^2)+b^3] - [3(a^2)(b) - 3a(b^2)]}
= (x+b) [(a+b)^3 - 3ab(a-b)]




2010-11-05 00:52:36 補充：
8) 如果未學 3次方的展開式....... (x+b)(a^3+b^3) 都已經係答案