因数 8(x-2)^3-27(x+1)^3

解

因数

8 (x - 2)^{3} - 27 (x + 1)^{3}

- (x + 7) (19 x^{2} - 4 x + 13)

解答ステップ

8 (x - 2)^{3} - 27 (x + 1)^{3}

8 (x - 2)^{3} - 27 (x + 1)^{3}

を書き換え

(2 (x - 2))^{3} - (3 (x + 1))^{3}

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

(2 (x - 2))^{3} - (3 (x + 1))^{3} = (2 (x - 2) - 3 (x + 1)) (2^{2} (x - 2)^{2} + 2 \cdot 3 (x + 1) (x - 2) + 3^{2} (x + 1)^{2})

= (2 (x - 2) - 3 (x + 1)) (3^{2} (x + 1)^{2} + 2 \cdot 3 (x + 1) (x - 2) + 2^{2} (x - 2)^{2})

改良

= (2 (x - 2) - 3 (x + 1)) (9 (x + 1)^{2} + 6 (x + 1) (x - 2) + 4 (x - 2)^{2})

因数

2 (x - 2) - 3 (x + 1) : - (x + 7)

= (- (x + 7)) (9 (x + 1)^{2} + 6 (x + 1) (x - 2) + 4 (x - 2)^{2})

括弧を削除する:

(- a) = - a

= - (x + 7) (9 (x + 1)^{2} + 6 (x + 1) (x - 2) + 4 (x - 2)^{2})

拡張

9 (x + 1)^{2} + 6 (x + 1) (x - 2) + 4 (x - 2)^{2} : 19 x^{2} - 4 x + 13

= - (x + 7) (19 x^{2} - 4 x + 13)