Question

初中因式分解数学竞赛题求解

1.x^2y^2+1-x^2-y^2+4xy 2.x^2-y^2+2x-4y-3 3.x^4+x^2-2ax-a+1 4.(x+y)^4+x^4+y^4 5.(a+b+c)^3-(a^3+b^3+c^3)
球详细解答过程

Answer 1

1.x^2y^2+1-x^2-y^2+4xy
=x²y²+2xy+1-x²+2xy-y²
=(xy+1)²-(x-y)²
=(xy+x-y+1)(xy-x+y+1)
2.x^2-y^2+2x-4y-3
=x²+2x+1-y²-4y-4
=(x+1)²-(y+2)²
=(x+1+y+2)(x+1-y-2)
=(x+y+3)(x-y-1)
3.x^4+x^2-2ax-a+1
=(x^4+2x^2+1)-(x^2+2ax+a^2)
=(x²+1)²-(x+a)²
=(x²-x-a+1)(x²+x+a+1)
4.(x+y)^4+x^4+y^4
=(x²+y²)²-2x²y²+(x+y)^4
=[(x+y)²-2xy]²-2x²y²+(x+y)^4
=2(x+y)^4-4xy(x+y)^4+2x²y²
=2[(x+y)²-xy]²
=2(x²+xy+y²)²
5.(a+b+c)^3-(a^3+b^3+c^3)
= 3(a²b+a²c+b²c+ab²+ac²+bc²+2abc)

Answer 2

2.原式=X^2+2X+1-（Y^2+4Y+4）=（X+1)^2-(Y+2)^2=(X+1+Y+2)(X+1-Y-2)

Answer 3

第一个：=(x^2y^2+1+2xy)-(x^2+y^2-2xy)
=(xy+1)^2-(x-y)^2
=（xy+1-x+y）(xy+1+x-y)
第二个：=(x^2+2x+1)-(y^2-4y+4)
=(x+1)^2-(y-2)^2
=(x+1+y-2)(x+1-y+2)
第三个不会做
=
第四个：
=2（x^4+y^4+2x^3y+3x^2y^2+2xy^3）
第五个
=3（a^2b+a^2c+b^2c+ab^2+ac^2+bc^2+2abc)

Answer 4

解：1、原式=(x^2y^2+2xy+1)-(x^2+y^2-2xy)=(xy+1)^2-(x-y)^2=(xy+1+x-y)(xy+1-x+y)
2、原式=（x^2+2x+1）-（y^2+4y+4）=（x+1)^2-(y+2)^2=(x+y+3)(x-y-1)