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因式分解 a^5b^{10}-c^{15}d^5

解答

因式分解

a^{5} b^{10} - c^{15} d^{5}

解答

(a b^{2} - c^{3} d) (a^{4} b^{8} + a^{3} b^{6} c^{3} d + a^{2} b^{4} c^{6} d^{2} + a b^{2} c^{9} d^{3} + c^{12} d^{4})

求解步骤

a^{5} b^{10} - c^{15} d^{5}

将

a^{5} b^{10}

改写为

(a b^{2})^{5}

将

c^{15} d^{5}

改写为

(c^{3} d)^{5}

= (a b^{2})^{5} - (c^{3} d)^{5}

使用因式分解法则

x^{n} - y^{n} = (x - y) (x^{n - 1} + x^{n - 2} y + \dots + x y^{n - 2} y^{n - 1})

(a b^{2})^{5} - (c^{3} d)^{5} = (a b^{2} - c^{3} d) ((a b^{2})^{4} + c^{3} d (a b^{2})^{3} + (c^{3} d)^{2} (a b^{2})^{2} + (c^{3} d)^{3} a b^{2} + (c^{3} d)^{4})

= (a b^{2} - c^{3} d) ((a b^{2})^{4} + c^{3} d (a b^{2})^{3} + (c^{3} d)^{2} (a b^{2})^{2} + (c^{3} d)^{3} a b^{2} + (c^{3} d)^{4})

化简

(a b^{2} - c^{3} d) ((a b^{2})^{4} + c^{3} d (a b^{2})^{3} + (c^{3} d)^{2} (a b^{2})^{2} + (c^{3} d)^{3} a b^{2} + (c^{3} d)^{4}) : (a b^{2} - c^{3} d) (a^{4} b^{8} + a^{3} b^{6} c^{3} d + a^{2} b^{4} c^{6} d^{2} + a b^{2} c^{9} d^{3} + c^{12} d^{4})

= (a b^{2} - c^{3} d) (a^{4} b^{8} + a^{3} b^{6} c^{3} d + a^{2} b^{4} c^{6} d^{2} + a b^{2} c^{9} d^{3} + c^{12} d^{4})

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