圆锥曲线中非对称代数式的处理方法

2019年浙江省高考数学卷圆锥曲线题的处理过程中,出现不能直接用韦达定理处理的非对称代数式.对于该问题,教师可借助对称式y1+y2,y1y2,|y1-y2|,y1/y2+y1/y2,化非对称代数式为对称式y1=1/2(y1+y2+|y1-y2|)y2=1/2[(y1+y2)-|y1-y2|].从而将任意的非对称代数式my1+ny2化为对称式my1+ny2=λ(y1+y2)+μ|y1-y2|),最终可以用韦达定理解决.

用“三法”求一元二次方程两根非对称代数式的值

<正>设一元二次方程ax2+bx+c=0的两根是x1、x2,要求不解方程,我们能够熟练地求出关于x1、x2的对称代数式(如x_1~2+x_2~2、x_1~3+x_2~3、1/x1+1/x2、(x1-x2)2、|x1-x2|等)的值.对含x1、x2的非对称代数式的值的求法,现举例介绍三种转化的方法:例设x1、x2中二次方程x2+x-3=0的两个根,那么x_1~3-4x_2~2+19的值是( )(1996年全国初中数学联赛)(A)- 4.(B)8.(C)6.(D)0.解法1:(配偶转化法):设A=x_1~3-4x_1~2+19,B=x_2~3-4x_1~2+19.∵x1、x2是方程x2+x-3=0的两根,∴x1+x2=-1,x1·x2=-3.

非对称式的求值方法

对于一元二次方程ax^2+bx+c=0(a≠0)，设它的两根为α、β，我们能够熟练地求出关于α、β的对称式如α^2+β^2、α^3+β^3、1/α+1/β、(α-β)^2、|α-β|等，对于α、β的非对称式的求值问题，关键是把非对称式转化为对称式，再利用与系数的关系进行求解．

Quantum Weyl symmetric polynomials and the center of quantum group U_q(sl_4)

Suppose that q is not a root of unity, it is proved in this paper that the center of the quantum group Uq(sl4) is isomorphic to a quotient algebra of polynomial algebra with four variables and one relation.

The Elements in Crystal Bases Corresponding to Exceptional Modules

According to the Ringel-Green theorem,the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group.Furthermore,its Drinfeld double can be identified with the whole quantum group,in which the BGP-reflection functors coincide with Lusztig's symmetries.It is first asserted that the elements corresponding to exceptional modules lie in the integral generic composition algebra,hence in the integral form of the quantum group.Then it is proved that these elements lie in the crystal basis up to a sign.Eventually,it is shown that the sign can be removed by the geometric method.The results hold for any type of Cartan datum.