因数 (x^2-1)^3-27

解

因数

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

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

解答ステップ

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

27

を書き換え

3^{3}

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

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

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

改良

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

因数

x^{2} - 4 : (x + 2) (x - 2)

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

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

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

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

拡張

(x + 1)^{2} (x - 1)^{2} + 3 (x + 1) (x - 1) + 9 : x^{4} + x^{2} + 7

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