因数 r^9-8(r^3-1)^3

解

因数

r^{9} - 8 (r^{3} - 1)^{3}

- (r^{3} - 2) (7 r^{6} - 10 r^{3} + 4)

解答ステップ

r^{9} - 8 (r^{3} - 1)^{3}

r^{9} - 8 (r^{3} - 1)^{3}

を書き換え

(r^{3})^{3} - (2 (r^{3} - 1))^{3}

= (r^{3})^{3} - (2 (r^{3} - 1))^{3}

立方数の差の公式を適用する:

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

(r^{3})^{3} - (2 (r^{3} - 1))^{3} = (r^{3} - 2 (r^{3} - 1)) (r^{6} + 2 r^{3} (r^{3} - 1) + 2^{2} (r^{3} - 1)^{2})

= (r^{3} - 2 (r^{3} - 1)) (r^{6} + 2 r^{3} (r^{3} - 1) + 2^{2} (r^{3} - 1)^{2})

改良

= (r^{3} - 2 (r^{3} - 1)) (r^{6} + 2 r^{3} (r^{3} - 1) + 4 (r^{3} - 1)^{2})

2 (r^{3} - 1) = 2 (r - 1) (r^{2} + r + 1)

= (r^{3} - 2 (r - 1) (r^{2} + r + 1)) (r^{6} + 2 r^{3} (r^{3} - 1) + 4 (r^{3} - 1)^{2})

2 r^{3} (r^{3} - 1) = 2 r^{3} (r - 1) (r^{2} + r + 1)

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

= (r^{3} - 2 (r - 1) (r^{2} + r + 1)) (r^{6} + 2 r^{3} (r - 1) (r^{2} + r + 1) + 4 (r - 1)^{2} (r^{2} + r + 1)^{2})

因数

r^{3} - 2 (r - 1) (r^{2} + r + 1) : - (r^{3} - 2)

= - (r^{3} - 2) (r^{6} + 2 r^{3} (r - 1) (r^{2} + r + 1) + 4 (r - 1)^{2} (r^{2} + r + 1)^{2})

拡張

r^{6} + 2 r^{3} (r - 1) (r^{2} + r + 1) + 4 (r - 1)^{2} (r^{2} + r + 1)^{2} : 7 r^{6} - 10 r^{3} + 4

= - (r^{3} - 2) (7 r^{6} - 10 r^{3} + 4)