Thread: Polynomial sub problem

for example, there is a function f(x)= x³+ax²+bx+c=0 with roots Q,W,E
then teachers told me that by substituting y=x² , then we can find out f(y)=0 s.t. it has roots Q²,W²,E²

but why such substition can help us to find out equations with root Q²,W²,E²? What are the reasons behind?
Thanks for your kind helpings

No, I don't think that works. We know that



Letting and yields a root that we already know...

oh, i mean by sub. y=x², then x³+ax²+bx+c = 0 becomes

x(x²+b) = -(a²+c)
x(y+b)=-(ay+c)
x²(y+b)²=(ay+c)²
y(y²+2by+b²) = (a²y²+2acy+c²)
y³+2by²+b²y = a²y² + 2acy + c²
y³+y²(2b-a²)+y(b²-2ac)-c² =0
then eqaution y³+y²(2b-a²)+y(b²-2ac)-c² =0, has required root, sorry for the misleading of using same function notication

You're way overthinking this. Also, your first step:



It should be

oh ya, sorry for my careless mistake... but why it works actually ><?