因式分解 (x+2y)^3-(2x-y)^3

解答

因式分解

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

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

求解步骤

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

使用立方差公式:

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

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

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

整理后得

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

乘开

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

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

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