標題:
唔識做功課6
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發問:
Factorize (x+y)^6 - (x-y)^6
最佳解答:
(x + y)? - (x - y)? = ( (x + y)2 )3 - ( (x - y)2 )3 = ( (x + y)2 - (x - y)2 ) ( (x + y)?+ (x + y)2 (x - y)2 + (x - y)?) = 4xy ( (x + y)?+ (x + y)2 (x - y)2 + (x - y)?) = 4xy ( (x + y)?+ 2(x + y)2 (x - y)2 + (x - y)? - (x + y)2 (x - y)2 ) = 4xy ( ( (x + y)2 + (x - y)2 )2 - (x + y)2 (x - y)2 ) = 4xy ( (2x2 + 2y2)2 - (x + y)2 (x - y)2 ) = 4xy ( (2x2 + 2y2)2 - (x2 - y2)2 ) = 4xy ( 2x2 + 2y2 - (x2 - y2) ) ( 2x2 + 2y2 + x2 - y2 ) = 4xy ( x2 + 3y2 ) ( 3x2 + y2 ) 2012-10-19 15:08:19 補充： Alternatively : (x + y)? - (x - y)? = ( (x + y)3 )2 - ( (x - y)3 )2 = ( (x + y)3 - (x - y)3 ) ( (x + y)3 + (x - y)3 ) = (x + y - (x - y)) ( (x + y)2 + (x + y)(x - y) + (x - y)2 ) (x + y + x - y)( (x + y)2 - (x + y)(x - y) + (x - y)2 ) 2012-10-19 15:08:25 補充： = 2y (x2 + 2xy + y2 + x2 - y2 + x2 - 2xy + y2) 2x (x2 + 2xy + y2 - (x2 - y2) + x2 - 2xy + y2) = 2y (3x2 + y2) 2x (x2 + 3y2) = 4xy ( x2 + 3y2 ) ( 3x2 + y2 ) 2012-10-19 15:12:27 補充： (x + y)?+ (x + y)2 (x - y)2 + (x - y)? 加 (x + y)2 (x - y)2 變 (x + y)?+ 2(x + y)2 (x - y)2 + (x - y)? 可拆成平方 ( (x + y)2 + (x - y)2 )2 因為加了 (x + y)2 (x - y)2 所以要減番 (x + y)2 (x - y)2 , 於是係 ( (x + y)2 + (x - y)2 )2 - (x + y)2 (x - y)2 又可用平方差分解了~
其他解答:
(x + y)? - (x - y)? = ( (x + y)2 )3 - ( (x - y)2 )3 = ( (x + y)2 - (x - y)2 ) ( (x + y)?+ (x + y)2 (x - y)2 + (x - y)?) = 4xy ( (x + y)?+ (x + y)2 (x - y)2 + (x - y)?) = 4xy ( (x + y)?+ 2(x + y)2 (x - y)2 + (x - y)? - (x + y)2 (x - y)2 ) = 4xy ( ( (x + y)2 + (x - y)2 )2 - (x + y)2 (x - y)2 ) = 4xy ( (2x2 + 2y2)2 - (x + y)2 (x - y)2 ) = 4xy ( (2x2 + 2y2)2 - (x2 - y2)2 ) = 4xy ( 2x2 + 2y2 - (x2 - y2) ) ( 2x2 + 2y2 + x2 - y2 ) = 4xy ( x2 + 3y2 ) ( 3x2 + y2 ) 先將6次變成a^3-b^3的公式 a^3-b^3 =(a+b)(a^2-ab+b^2) ((a)^2)^3=a^2*3 之後再抽共有因子 這D題有好多種方法計|||||唔知你有冇學過呢種既因式解既方法,利用nCr (x+y)^6=x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6 (x-y)^6=x^6 - 6x^5y + 15x^4y^2 - 20x^3y^3 + 15x^2y^4 - 6xy^5 + y^6 (x+y)^6 - (x-y)^6 =12x^5y+40x^3y^3 +12xy^5|||||我唔明第四個等於係如何得出來,可否再解釋,感激！感激！
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