①x²+y²-2xy-10x+10y+16;

题目

①x²+y²-2xy-10x+10y+16;
②2x²+3xy-2y²-x+8y-6;
③x²-2xy-3y²+3x-5y+2;
④6x²-7xy-3y²-xz+7yz-2z².

答案

①x²+y²-2xy-10x+10y+16;
=(x-y)²-10(x-y)+16
=(x-y-2)(x-y-8)
②2x²+3xy-2y²-x+8y-6;
因为 2x²+3xy-2y²=(2x-y)(x+2y)
所以设 2x²+3xy-2y²-x+8y-6=(2x-y+a)(x+2y+b)
=2x²+3xy-2y²+ax+2ay+2bx-by+ab
=2x²+3xy-2y²+(a+2b)x+(2a-b)y+ab
所以 a+2b=-1,2a-b=8,ab=-6
解得 a=3,b=-2
所以 2x²+3xy-2y²-x+8y-6=(2x-y+3)(x+2y-2)
③x²-2xy-3y²+3x-5y+2;
因为 x²-2xy-3y²=(x-3y)(x+y)
所以设 x²-2xy-3y²+3x-5y+2=(x-3y+a)(x+y+b)
=x²-2xy-3y²+ax+ay+bx-3by+ab
=x²-2xy-3y²+(a+b)x+(a-3b)y+ab
所以 a+b=3,a-3b=-5,ab=2
解得 a=1,b=2
所以 x²-2xy-3y²+3x-5y+2=(x-3y+1)(x+y+2)
④6x²-7xy-3y²-xz+7yz-2z².
因为 6x²-7xy-3y²=(2x-3y)(3x+y)
所以设 6x²-7xy-3y²-xz+7yz-2z²=(2x-3y+az)(3x+y+bz)
=6x²-7xy-3y²+3axz+ayz+2bxz-3byz+abz²
=6x²-7xy-3y²+(3a+2b)xz+(a-3b)yz+(ab)z²
所以 3a+2b=-1,a-3b=7,ab=-2
解得 a=1,b=-2
所以 6x²-7xy-3y²-xz+7yz-2z²=(2x-3y+z)(3x+y-2z)