因数 x^4+4x^3+4x^2-y^2

解

因数

x^{4} + 4 x^{3} + 4 x^{2} - y^{2}

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

解答ステップ

x^{4} + 4 x^{3} + 4 x^{2} - y^{2}

因数

x^{4} + 4 x^{3} + 4 x^{2} : x^{2} (x + 2)^{2}

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

指数の規則を適用する:

a^{n} b^{n} = (a \cdot b)^{n}

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

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

2乗の差の公式を適用する:

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

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

= (x (x + 2) + y) (x (x + 2) - y)

簡素化

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

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