分解因式.1.（x^2+y^2）(x^2-xy+y^2)-2x^2y^22.x^3+6x^2+11x+63.a^2-4ab+3b^2+2bc-c^24.a^4+a^2b^2+b^45.若x^2+2x+5是x^4+mx^2+n的一个因式,求m,n的值.2题.答案是(x+3)(x+2)(x+1)3题.答案是(a-b-c)(a-3b+c)5题.答案是m=6n=251.4题解决

Question

分解因式.
1.（x^2+y^2）(x^2-xy+y^2)-2x^2y^2
2.x^3+6x^2+11x+6
3.a^2-4ab+3b^2+2bc-c^2
4.a^4+a^2b^2+b^4
5.若x^2+2x+5是x^4+mx^2+n的一个因式,求m,n的值.
2题.答案是(x+3)(x+2)(x+1)
3题.答案是(a-b-c)(a-3b+c)
5题.答案是m=6 n=25
1.4题解决了,

Answer 1

1 =x^4-x^3y+2x^2y^2-xy^3+y^4-2x^2y^2
=x^4-x^3y-xy^3+y^4
=x^3(x-y)-y^3(x-y)
=(x-y)(x^3-y^3)
=(x-y)^2(x^2+xy+y^2)
2 =(x^3+5x^2+6x)+(x^2+5x+6)
=x(x+2)(x+3)+(x+2)(x+3)
=(x+1)(x+2)(x+3)
3 =(a-2b)^2-(b^2-2bc+c^2)
=(a-2b)^2-(b-c)^2
=(a-2b+b-c)(a-2b-b+c)
=(a-b-c)(a-3b+c)
4 =a^4+2a^2b^2+b^4-a^2b^2
=(a^2+b^2)^2-a^2b^2
=(a^2+b^2+ab)(a^2+b^2-ab)
5 x^4+mx^2+n=(x^2+2x+5)乘以一个未知的式子
假设式子为 ax^2+bx+c
则
(ax^2+bx+c)(x^2+2x+5)=ax^4+2ax^3+5ax^2+bx^3+2bx^2+5bx+cx^2+2cx+5c
=ax^4+(2a+b)x^3+(5a+2b+c)x^2+(5b+2c)x+5c
=x^4+mx^2+n
所以
a=1 2a+b=0 5a+2b+c=m 5b+2c=0 5c=n
即 a=1 b=-2 c=5
m=6 n=25