因数 32(x+1)^5-243y^{10}

解

因数

32 (x + 1)^{5} - 243 y^{10}

(2 (x + 1) - 3 y^{2}) (16 (x + 1)^{4} + 24 y^{2} (x + 1)^{3} + 36 y^{4} (x + 1)^{2} + 54 y^{6} (x + 1) + 81 y^{8})

解答ステップ

32 (x + 1)^{5} - 243 y^{10}

32 (x + 1)^{5}

を書き換え

(2 (x + 1))^{5}

243 y^{10}

を書き換え

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

= (2 (x + 1))^{5} - (3 y^{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 (x + 1))^{5} - (3 y^{2})^{5} = (2 (x + 1) - 3 y^{2}) ((2 (x + 1))^{4} + 3 y^{2} (2 (x + 1))^{3} + (3 y^{2})^{2} (2 (x + 1))^{2} + (3 y^{2})^{3} 2 (x + 1) + (3 y^{2})^{4})

= (2 (x + 1) - 3 y^{2}) ((2 (x + 1))^{4} + 3 y^{2} (2 (x + 1))^{3} + (3 y^{2})^{2} (2 (x + 1))^{2} + (3 y^{2})^{3} 2 (x + 1) + (3 y^{2})^{4})

簡素化

(2 (x + 1) - 3 y^{2}) ((2 (x + 1))^{4} + 3 y^{2} (2 (x + 1))^{3} + (3 y^{2})^{2} (2 (x + 1))^{2} + (3 y^{2})^{3} \cdot 2 (x + 1) + (3 y^{2})^{4}) : (2 (x + 1) - 3 y^{2}) (16 (x + 1)^{4} + 24 y^{2} (x + 1)^{3} + 36 y^{4} (x + 1)^{2} + 54 y^{6} (x + 1) + 81 y^{8})

= (2 (x + 1) - 3 y^{2}) (16 (x + 1)^{4} + 24 y^{2} (x + 1)^{3} + 36 y^{4} (x + 1)^{2} + 54 y^{6} (x + 1) + 81 y^{8})