因数 (a^2-1)^3-27

解

因数

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

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

解答ステップ

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

27

を書き換え

3^{3}

= (a^{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})

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

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

改良

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

因数

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

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

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

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

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

拡張

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

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