Math 問題(20分)

(x-2)^3+(2x+3)^3
= (x-2+2x+3) [ (x-2)^2 - (x-2)(2X+3) + (2x+3)^2 ]
= (3x+1) [ (x^2-4x+4 - (2x^2+3x-4x-6) + (4x^2+12x+9) ]
= (3x+1) (3x^2+9x+11)

Also,
(x-2)^2 是 = (x-2)(x-2) not (x-2)(x+2)
平常的應該是(x-2)^2 = (x-2)(x-2)

(a-b)^2 = ( a^2 - 2ab + b^2 )
a^2 - b^2 = ( a - b )^2　　　　　呢到記得背熟佢！好有用架

step應該係咁（我自己計唔知岩唔岩＝＝）

[(x - 2)(x - 2 )^2] + [(2x + 3)(2x + 3)^2]

= [(x -2)(x^2 - 4x + 4)] + [(2x + 3)(4x^2 +12x + 9)]

= [x(x^2 - 4x +4) - 2(x^2 - 4x + 4)] + [2x(4x^2 +12x + 9) + 3(4x^2 +12x +9)]

= (x^3 - 4x^2 + 4x - 2x^2 + 8x - 8) + (8x^3 + 24x^2 +18x + 12x^2 + 36x + 27)

= (x^3 - 6x^2 + 12x - 8) + (8x^3 + 36x^2 + 54x + 27)

= 9x^3 + 30x^2 + 66x +19 //

(x-2)^3+(2x+3)^3
= (x-2+2x+3) [ (x-2)^2 - (x-2)(2X+3) + (2x+3)^2 ] ...... 因為 a^3+b^3 = (a+b)(a^2-ab+b^2)
= (3x + 1) [(x^2 - 4x + 4) - (2x^2 - x - 6) + (4x^2 + 12x + 9)] ..... 將 [] 內的 3 組 ( ) 及 ^2 乘開
= (3x + 1) (x^2 - 4x + 4 - 2x^2 + x + 6 + 4x^2 + 12x + 9)
= (3x + 1) (3x^2 + 9x + 19)
= 3x * (3x^2 + 9x + 19) + 1 * (3x^2 + 9x + 19)
= 9x^3 + 27x^2 + 57x + 3x^2 + 9x + 19
= 9x^3 + 30x^2 + 66x + 19


Note:
x^2 - 2^2 = (x-2)(x+2) ...... 2 次方是個別 (x) 的2次方和 (2) 的2次方
(x-2)^2 = (x-2)(x-2) = x^2 - 4x + 4 ...... 2 次方包括整個 (x-2)
同樣 (x+2)^2 = (x+2)(x+2) = x^2 + 4x + 4 ...... 2 次方包括整個 (x+2)
所以 (a-b)^2 跟 a^2-b^2 是不同的, 要小心留意括號.