因数 (x+y)^3-8(x-y)^3

解

因数

(x + y)^{3} - 8 (x - y)^{3}

(- x + 3 y) (7 x^{2} - 6 x y + 3 y^{2})

解答ステップ

(x + y)^{3} - 8 (x - y)^{3}

(x + y)^{3} - 8 (x - y)^{3}

を書き換え

(x + y)^{3} - (2 (x - y))^{3}

= (x + y)^{3} - (2 (x - y))^{3}

立方数の差の公式を適用する:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

(x + y)^{3} - (2 (x - y))^{3} = ((x + y) - 2 (x - y)) ((x + y)^{2} + 2 (x - y) (x + y) + 2^{2} (x - y)^{2})

= ((x + y) - 2 (x - y)) (2^{2} (x - y)^{2} + 2 (x - y) (x + y) + (x + y)^{2})

改良

= (x + y - 2 (x - y)) ((x + y)^{2} + 2 (x - y) (x + y) + 4 (x - y)^{2})

拡張

x + y - 2 (x - y) : - x + 3 y

= (3 y - x) (2 (x - y) (x + y) + (x + y)^{2} + 4 (x - y)^{2})

拡張

4 (x - y)^{2} + 2 (x - y) (x + y) + (x + y)^{2} : 7 x^{2} - 6 x y + 3 y^{2}

= (3 y - x) (7 x^{2} - 6 x y + 3 y^{2})