因数 32m^{15}-243n^{10}

解

因数

32 m^{15} - 243 n^{10}

(2 m^{3} - 3 n^{2}) (16 m^{12} + 24 m^{9} n^{2} + 36 m^{6} n^{4} + 54 m^{3} n^{6} + 81 n^{8})

解答ステップ

32 m^{15} - 243 n^{10}

32 m^{15}

を書き換え

(2 m^{3})^{5}

243 n^{10}

を書き換え

(3 n^{2})^{5}

= (2 m^{3})^{5} - (3 n^{2})^{5}

因数分解の規則を適用する:

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

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

= (2 m^{3} - 3 n^{2}) ((2 m^{3})^{4} + (2 m^{3})^{3} 3 n^{2} + (2 m^{3})^{2} (3 n^{2})^{2} + 2 m^{3} (3 n^{2})^{3} + (3 n^{2})^{4})

簡素化

(2 m^{3} - 3 n^{2}) ((2 m^{3})^{4} + (2 m^{3})^{3} \cdot 3 n^{2} + (2 m^{3})^{2} (3 n^{2})^{2} + 2 m^{3} (3 n^{2})^{3} + (3 n^{2})^{4}) : (2 m^{3} - 3 n^{2}) (16 m^{12} + 24 m^{9} n^{2} + 36 m^{6} n^{4} + 54 m^{3} n^{6} + 81 n^{8})

= (2 m^{3} - 3 n^{2}) (16 m^{12} + 24 m^{9} n^{2} + 36 m^{6} n^{4} + 54 m^{3} n^{6} + 81 n^{8})