因式分解 (x-4)^3-(2x+5)^3

解答

因式分解

(x - 4)^{3} - (2 x + 5)^{3}

(- x - 9) (7 x^{2} + 9 x + 21)

求解步骤

(x - 4)^{3} - (2 x + 5)^{3}

使用立方差公式:

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

(x - 4)^{3} - (2 x + 5)^{3} = ((x - 4) - (2 x + 5)) ((x - 4)^{2} + (x - 4) (2 x + 5) + (2 x + 5)^{2})

= ((x - 4) - (2 x + 5)) ((x - 4)^{2} + (x - 4) (2 x + 5) + (2 x + 5)^{2})

整理后得

= (- x - 9) ((2 x + 5)^{2} + (x - 4) (2 x + 5) + (x - 4)^{2})

乘开

(x - 4)^{2} + (x - 4) (2 x + 5) + (2 x + 5)^{2} : 7 x^{2} + 9 x + 21

= (- x - 9) (7 x^{2} + 9 x + 21)

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