Question

分解因式两道。

（1）x^4+y^4+z^4-2x^2y^2+2x^2z^2-2y^2z^2
(2)(x^2+3x)^2-(2x+6)^2
要过程，谢谢、。

Answer 1

1)
x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2
=x^4+y^4+z^4+2x^2y^2-2x^2z^2-2y^2z^2-4x^2y^2
=(x^2+y^2-z^2)^2-4x^2y^2
=(x^2+y^2-z^2+2xy)(x^2+y^2-z^2-2xy)
=[(x+y)^2-z^2][(x-y)^2-z^2]
=(x+y+z)(x+y-z)(x-y+z)(x-y-z)

2)
平方差公式
(x^2+3x)^2-(2x+6)^2
=（x^2+3x-2x-6）(x^2+3x+2x+6)
=(x^2+x-6)(x^2+5x+6)
=(x+3)(x-2)(x+3)(x+2)
=(x+3)^2(x-2)(x+2)

Answer 2

x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2
=x^4+y^4+z^4+2x^2y^2-2x^2z^2-2y^2z^2-4x^2y^2
=(x^2+y^2-z^2)^2-4x^2y^2
=(x^2+y^2-z^2+2xy)(x^2+y^2-z^2-2xy)
=[(x+y)^2-z^2][(x-y)^2-z^2]
=(x+y+z)(x+y-z)(x-y+z)(x-y-z)

(x^2+3x)^2-(2x+6)^2
＝（x^2+3x+2x+6)(x^2+3x-2x-6)
=(x^2+5x+6)(x^2+x-6)
=(x+2)(x+3)(x+2)(x-3)
=(x+2)^2(x+3)(x-3)

Answer 3

1）x^4+y^4+z^4-2x^2y^2+2x^2z^2-2y^2z^2
=(x²+y²-z²)²-4x²y²
=(x²+y²-z²+2xy)(x²+y²-z^2-2xy)
=[(x+y)²-z²][(x-y)²-z²]
=(x+y+z)(x+y-z)(x-y+z)(x-y-z)

2)(x^2+3x)^2-(2x+6)^2
=(x²+3x)²-4(x+3)²
=x²(x+3x)²-4(x+3)²
=(x+3)²(x²-4)
=(x+3)²(x+2)(x-2)