因数 x^{12}-1

解

因数

x^{12} - 1

(x^{2} + 1) (x + 1) (x - 1) (x^{2} + x + 1) (x^{2} - x + 1) (x^{4} - x^{2} + 1)

解答ステップ

x^{12} - 1

指数の規則を適用する

= (x^{4})^{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})

(x^{4})^{3} - 1^{3} = (x^{4} - 1) (x^{8} + x^{4} + 1)

= (x^{4} - 1) (x^{8} + x^{4} + 1)

因数

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

= (x^{2} + 1) (x + 1) (x - 1) (x^{8} + x^{4} + 1)

因数

x^{8} + x^{4} + 1 : (x^{2} + 1 + x) (x^{2} + 1 - x) (x^{4} + 1 - x^{2})

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

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