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受欢迎的代数 >

化简 (x+y+z)^3-(x-y-z)^3-6(x+y+z)(x-y-z)(y+z)

解答

化简

(x + y + z)^{3} - (x - y - z)^{3} - 6 (x + y + z) (x - y - z) (y + z)

解答

24 y z^{2} + 8 z^{3} + 24 y^{2} z + 8 y^{3}

求解步骤

(x + y + z)^{3} - (x - y - z)^{3} - 6 (x + y + z) (x - y - z) (y + z)

展开

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

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

展开

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

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

展开

- 6 (x + y + z) (x - y - z) (y + z) : - 6 x^{2} y + 18 y^{2} z + 18 y z^{2} + 6 y^{3} + 6 z^{3} - 6 x^{2} z

= x^{3} + 3 x^{2} y + 3 x^{2} z + 3 x z^{2} + 3 y z^{2} + 3 x y^{2} + z^{3} + 3 y^{2} z + 6 x yz + y^{3} - x^{3} + 3 x^{2} y + 3 x^{2} z - 3 x z^{2} + 3 y z^{2} - 3 x y^{2} + z^{3} + 3 y^{2} z - 6 x yz + y^{3} - 6 x^{2} y + 18 y^{2} z + 18 y z^{2} + 6 y^{3} + 6 z^{3} - 6 x^{2} z

对同类项分组

= x^{3} - x^{3} + 3 x^{2} y + 3 x^{2} y - 6 x^{2} y + 3 x^{2} z + 3 x^{2} z - 6 x^{2} z + 3 x z^{2} - 3 x z^{2} + 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

x^{3} - x^{3} = 0

= 3 x^{2} y + 3 x^{2} y - 6 x^{2} y + 3 x^{2} z + 3 x^{2} z - 6 x^{2} z + 3 x z^{2} - 3 x z^{2} + 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

3 x^{2} y + 3 x^{2} y - 6 x^{2} y = 0

= 3 x^{2} z + 3 x^{2} z - 6 x^{2} z + 3 x z^{2} - 3 x z^{2} + 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

3 x^{2} z + 3 x^{2} z - 6 x^{2} z = 0

= 3 x z^{2} - 3 x z^{2} + 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

3 x z^{2} - 3 x z^{2} = 0

= 3 y z^{2} + 3 y z^{2} + 18 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

3 y z^{2} + 3 y z^{2} + 18 y z^{2} = 24 y z^{2}

= 24 y z^{2} + 3 x y^{2} - 3 x y^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

3 x y^{2} - 3 x y^{2} = 0

= 24 y z^{2} + z^{3} + 6 z^{3} + z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

z^{3} + 6 z^{3} + z^{3} = 8 z^{3}

= 24 y z^{2} + 8 z^{3} + 3 y^{2} z + 3 y^{2} z + 18 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

3 y^{2} z + 3 y^{2} z + 18 y^{2} z = 24 y^{2} z

= 24 y z^{2} + 8 z^{3} + 24 y^{2} z + 6 x yz - 6 x yz + y^{3} + 6 y^{3} + y^{3}

同类项相加:

6 x yz - 6 x yz = 0

= 24 y z^{2} + 8 z^{3} + 24 y^{2} z + y^{3} + 6 y^{3} + y^{3}

同类项相加:

y^{3} + 6 y^{3} + y^{3} = 8 y^{3}

= 24 y z^{2} + 8 z^{3} + 24 y^{2} z + 8 y^{3}

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