因数 (x+y+z)^3-x^3-y^3-z^3

解

因数

(x + y + z)^{3} - x^{3} - y^{3} - z^{3}

3 (x^{2} y + x^{2} z + x y^{2} + x z^{2} + 2 x yz + y z^{2} + y^{2} z)

解答ステップ

(x + y + z)^{3} - x^{3} - y^{3} - z^{3}

(x + y + z)^{3} = (x + y + z) (x + y + z) (x + y + z)

= (x + y + z) (x + y + z) (x + y + z) - x^{3} - y^{3} - z^{3}

拡張

(x + y + z) (x + y + z) (x + y + z) : x^{3} + 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + y^{3} + z^{3} + 3 y z^{2} + 3 y^{2} z

= x^{3} + 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + y^{3} + z^{3} + 3 y z^{2} + 3 y^{2} z - x^{3} - y^{3} - z^{3}

簡素化

x^{3} + 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + y^{3} + z^{3} + 3 y z^{2} + 3 y^{2} z - x^{3} - y^{3} - z^{3} : 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + 3 y z^{2} + 3 y^{2} z

= 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 6 x yz + 3 y z^{2} + 3 y^{2} z

書き換え

= 3 x^{2} y + 3 x^{2} z + 3 x y^{2} + 3 x z^{2} + 3 \cdot 2 x yz + 3 y z^{2} + 3 y^{2} z

共通項をくくり出す

3

= 3 (x^{2} y + x^{2} z + x y^{2} + x z^{2} + 2 x yz + y z^{2} + y^{2} z)