Find polynoms, with as least as power possible

[tiny-tim]

I tried only A(x)=P and B(x)=Q and it didn't work, because of

$$P(\alpha ax^n + \alpha b) + Q(\beta cx^n + \beta d)$$

so

$$\frac{\alpha}{0}=\frac{0}{\beta}$$ and $$\frac{\beta}{0}=\frac{0}{d}$$

Exactly!

can you tell me please, or start writing that formula?

No! It's a waste of time - as I said, it'll always be easier to do it one step at a time!

 

[Physicsissuef]

Just start, please. I will continue... I just want to know that. I wouldn't use it...

 

[tiny-tim]

No! You start, if you like, and I'll comment …

 

[Physicsissuef]

The start, will be like this?
$$(ax^n+bx^n^-^1+cx^n^-^2+...+ex^n^-^f)(\alpha P+\beta Q)+(gx^n+hx^n^-^1+ix^n^-^2+...+kx^n^-^w)(\gamma P +\delta Q)$$

 

[tiny-tim]

Go on …

 

[Physicsissuef]

$$P(x^n(a\alpha + g\gamma) + x^n^-^1(b\alpha + h\gamma) + x^n^-^2(c\alpha + i\gamma) +...+ \alpha ex^n^-^f + \gamma kx^n^-^w)$$

like this? Should I go on for Q?

 

[tiny-tim]

… how did we get from A and B to U and V … ?

I've no idea - I haven't worked out what your plan is !

Perhaps it would be better to go back to the original example, and analyse how we got from A and B to U and V (that would be two steps in one, which is what you're looking for), and then sort out a plan of campaign from that?

 

[Physicsissuef]

I don't know what you mean... Sorry...

 

[tiny-tim]

Oh … I'm getting my letters mixed up!

I forgot we got through so many letters …

I meant "from A and B to G and I".

 

[Physicsissuef]

A and B to G and I from the last https://www.physicsforums.com/showpost.php?p=1650515&postcount=102"
A and B first step, and G and I are from the last step, how will I jump "so high" ?

 

[tiny-tim]

I don't know … but I thought that's what you were aiming for, to do two steps (A and B to P and Q, then P and Q to G and I) in one go??

 

[Physicsissuef]

And how will I do that?

 

[tiny-tim]

I don't know!

 

[Physicsissuef]

c'mooon :) just start... I have no idea... Just let's finish this once forever

 

[tiny-tim]

Noooooooo! ​

 

[Physicsissuef]

$$(ax^n\,+\,...\,+\,b)(\alpha P\,+\,\beta Q)\,+\,(cx^n\,+\,...\,+\,d)(\gamma P\,+\,\delta Q)$$$$=\,P((...+\,(\alpha b\,+\,\gamma d))\,+\,Q((\beta a\,+\,\delta c)x^n\,+\,...)\,;$$

$$P=xR$$

btw- how will I know P or Q need xR in the general formula? Is it good now?

 

[Physicsissuef]

tiny-tim, where are you? Can you help me please go on?