Chapter 1 Solving algebraic equations
Cubic equations
To solve the equation 
, let 
to transform the equation to the form
We have the identity:
%5E3-3ab(a%2Bb)-(a%5E3%2Bb%5E3)%3D0)
thus if we find 
such that %20%5Ctag%7B2%7D)
then 
is a root of the equation(1).
By (2) we can see that 
are roots of the equation
Quadratic equations
To solve the equation 
, which can be written as ![(X^2+\frac{m}{2}X+\frac{\lambda}{2})^2-[(\frac{m^2}{4}+\lambda-p)X^2+(\frac{m\lambda}{2}-q)X+(\frac{\lambda^2}{4}-r)]=0](https://math.jianshu.com/math?formula=(X%5E2%2B%5Cfrac%7Bm%7D%7B2%7DX%2B%5Cfrac%7B%5Clambda%7D%7B2%7D)%5E2-%5B(%5Cfrac%7Bm%5E2%7D%7B4%7D%2B%5Clambda-p)X%5E2%2B(%5Cfrac%7Bm%5Clambda%7D%7B2%7D-q)X%2B(%5Cfrac%7B%5Clambda%5E2%7D%7B4%7D-r)%5D%3D0)
find a parameter 
such that the latter polynomial is a square of a polynomial of degree 1,then the quadratic polynomial is a product of two polymials of degree 2.
To find the , we need to solve the equation
%5E2-(%5Cfrac%7Bm%5E2%7D%7B4%7D%2B%5Clambda-p)(%5Cfrac%7B%5Clambda%5E2%7D%7B4%7D-r)%3D0)
Its a cubic equation.
