因式分解 (p+q)^3-(p-q)^3

解答

因式分解

(p + q)^{3} - (p - q)^{3}

2 q (3 p^{2} + q^{2})

求解步骤

(p + q)^{3} - (p - q)^{3}

使用立方差公式:

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

(p + q)^{3} - (p - q)^{3} = ((p + q) - (p - q)) ((p + q)^{2} + (p - q) (p + q) + (p - q)^{2})

= ((p + q) - (p - q)) ((p - q)^{2} + (p - q) (p + q) + (p + q)^{2})

整理后得

= 2 q ((p + q)^{2} + (p - q) (p + q) + (p - q)^{2})

乘开

(p + q)^{2} + (p - q) (p + q) + (p - q)^{2} : 3 p^{2} + q^{2}

= 2 q (3 p^{2} + q^{2})

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