Thread: Polynomial sub problem

1. 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

2. Re: Polynomial sub problem

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

$f(y) = f(x^2) = (x^2)^3 + a(x^2)^2 + bx^2 + c$

Letting $x = \sqrt{Q}$ and $y = Q$ yields a root that we already know...

3. Re: Polynomial sub problem

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

4. Re: Polynomial sub problem

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

$x(x^2 + b) = -(a^2 + c)$

It should be

$x(x^2 + b) = -(ax^2 + c)$

5. Re: Polynomial sub problem

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