Find polynoms, with as least as power possible

[Physicsissuef]

I understand everything, just this part:
A = dP + cQ, B = -(bP + aQ);
How did u find it?

So actually our mission is to contract the polynom (the 1st and the last element), right?

 

[tiny-tim]

… your mission, if you choose to accept it, is …

So actually our mission is to contract the polynom (the 1st and the last element), right?

That's right!

I understand everything, just this part:
A = dP + cQ, B = -(bP + aQ);
How did u find it?

Because, since we know:

$$\frac{\alpha}{\gamma}\,=\,\frac{-d}{b}\,;\,\frac{\beta}{\delta}\,=\,\frac{-c}{a}\,;$$​

the obvIous thing is to put:

$$\alpha\,=\,-d\,,\,\beta\,=\,-c\,,\,\gamma\,=\,b\,,\,\delta\,=\,a\,,$$​

though I instead decided to multiply them all by minus-one (that's ok because it leaves the ratios, in the previous line, the same):

$$\alpha\,=\,d\,,\,\beta\,=\,c\,,\,\gamma\,=\,-b\,,\,\delta\,=\,-a\,,$$​

(because that made A positive, which I think looks neater ),

which is the same as A = dP + cQ, B = -(bP + aQ). ​

 

[Physicsissuef]

You're very intelligent. And what is one power less than $x^4-x^2$, is it $x^3-x$?

 

[tiny-tim]

… fourth-order in x, but only second-order in x^2 …

And what is one power less than $x^4-x^2$, is it $x^3-x$?

Ah! You're referring to:

$$x^2(x^2\,-\,1)U\,+\,(x^2\,+\,1)V\,=\,2x^2\,-\,4\,;$$

I tried to substitute $U=ax^2+bx+c$ and $V=dx^2+ex+f$

and it didn't worked.

I agree that $\LARGE x^4\,-\,x^2$ is fourth-order in x, so we should normally use cubics for the e f g etc method - but I treated that equation as an equation in x^2 rather than x (because there's no odd powers of x in it), and it's only second-order in x^2, so m = 1, and I could use the linear-in-x^2 ax + b, cx + d.

 

[Physicsissuef]

Ok, I understand. I have one question more. You write this:
$$=\,P((...+\,(\alpha b\,+\,\gamma d))\,+\,Q((\beta a\,+\,\delta c)x^n\,+\,...)\,;$$
Which is absolute genius. I want to ask you, can we contract the polynom much more, using $x^n^-^1$?

 

[tiny-tim]

… remainders … ?

I want to ask you, can we contract the polynom much more, using $x^n^-^1$?

Hi Physicsissuef

If you mean, can we reduce from x^n to x^n-1, and then reduce again from x^n-1 to x^n-2, and so on, then yes … in fact, it's what you just did with you own example.

But please remember that, although this has been good practice at manipulating polynomials, it's rather slow, and easy to make mistakes in, and the a b c d e f g h j k etc method right from the beginning is probably better in nearly all cases!

And I still suspsect that there's some trick involving remainders that we haven't spotted …

 

[Physicsissuef]

Hi Physicsissuef

If you mean, can we reduce from x^n to x^n-1, and then reduce again from x^n-1 to x^n-2, and so on, then yes … in fact, it's what you just did with you own example.

But please remember that, although this has been good practice at manipulating polynomials, it's rather slow, and easy to make mistakes in, and the a b c d e f g h j k etc method right from the beginning is probably better in nearly all cases!

And I still suspsect that there's some trick involving remainders that we haven't spotted …

-------------------------------------------------------------

I thought like this:

$$(ax^n+bx^n^-^1+cx^n^-^2+dx^n^-^3+ex^n^-^f)(\alpha P+\beta Q)+(gx^n+hx^n^-^1+ix^n^-^2+jx^n^-^3+kx^n^-^w)(\gamma P +\delta Q)$$

To find out some forumula
So we can "jump" through 2-3 steps (the steps that we went all over again).

-------------------------------------------------------------------

Other question.
Can I use just $$\alpha P$$ instead of $$\alpha P+\beta Q$$ and $$\gamma P +\delta Q$$?

----------------------------------------------------------------------

Another question.

What will happen if I have something like this?
$$(x^4-2x^3-4x^2+6x+1)A(x)+(x^3-5x-3)B(x)=x-2$$

Another question.

---------------------------------------------------------------------

What will happen if d=c=1 and a=b=-1
Lets say the polynom is:
$$(-x^3+2x^2+3x-1)A(x) + (x^3-5x^2+7x+1)B(x)=x^2-4x$$

$$A(x)=B(x)$$

?

 

[tiny-tim]

I thought like this:
$$(ax^n+bx^n^-^1+cx^n^-^2+dx^n^-^3+ex^n^-^f)(\alpha P+\beta Q)+(gx^n+hx^n^-^1+ix^n^-^2+jx^n^-^3+kx^n^-^w)(\gamma P +\delta Q)$$
To find out some forumula so we can "jump" through 2-3 steps (the steps that we went all over again).

There obviously is a formula for doing 2 steps together - but it would be much more complicated, and doing the steps separately is easier and has less risk of mistakes!

Can I use just $$\alpha P$$ instead of $$\alpha P+\beta Q$$ and $$\gamma P +\delta Q$$?
What will happen if I have something like this?
$$(x^4-2x^3-4x^2+6x+1)A(x)+(x^3-5x-3)B(x)=x-2$$

Well, that's a = 1, b = 1, c = 0, d = -3, which is ok!

What will happen if d=c=1 and a=b=-1
Lets say the polynom is:
$$(-x^3+2x^2+3x-1)A(x) + (x^3-5x^2+7x+1)B(x)=x^2-4x$$

$$A(x)=B(x)$$ ?

Now, that is a very good question!

You're right - it doesn't work if the ratio a/b is the same as c/d (you gave a = -c, b = -d; but it's just as bad if a = c, b = d, or a = 3c, b = 3d …).

And we certainly can't put A(x) = B(x) (or any multiple of B(x))!

So we'd have to go back to the a b c d e f g etc method (which, I repeat, is probably the best method anyway)

 

[Physicsissuef]

I think in some ways work.

$$(x^4-2x^3-4x^2+6x+1)A(x)+(x^3-5x-3)B(x)=x-2$$

In this case I can not divide by x , because I have x-2 on the right hand. In this case also will not work?

 

[tiny-tim]

Yes, of course: we only used this method because the right-hand side had no units (only positive powers of x).

If the right-hand side has units, as in your present example, then eliminating units from the left-hand side and dividing by x (the first step) would be useless!

(btw, I did a google search on … something … and I now know who StMartin is, and what he's doing! … the secret is safe with me! )​

 

[Physicsissuef]

Yes, of course: we only used this method because the right-hand side had no units (only positive powers of x).

If the right-hand side has units, as in your present example, then eliminating units from the left-hand side and dividing by x (the first step) would be useless!

(btw, I did a google search on … something … and I now know who StMartin is, and what he's doing! … the secret is safe with me! )​

Hmmm... I can't see any secret Why secret?
And I want to ask you something else.

Can I use just $$A(x) = \alpha P$$ and $$B(x) = \beta Q$$ instead of
$$A(x) = \alpha P+\beta Q$$ and $$B(x) = \gamma P +\delta Q$$?

 

[tiny-tim]

Can I use just $$A(x) = \alpha P$$ and $$B(x) = \beta Q$$ instead of
$$A(x) = \alpha P+\beta Q$$ and $$B(x) = \gamma P +\delta Q$$?

Well, we can make any substitution we like!

The resulting new equation will still be valid - but will it be any easier to solve?

I suppose $$A(x) = \alpha P$$ and $$B(x) = \beta Q$$ might sometimes be useful - were you thinking of a particular example?

 

[Physicsissuef]

No, I don't. But I think it is not useful, because, often can happen A(x)=B(x).

And if we want to "jump" 2 or 3 steps or maybe more, do we must know the power of the polynom (ex. $x^3$, or $x^4$) ??
Like this:
$$(ax^4+bx^3+cx^n^2+dx^1+e)(\alpha P+\beta Q)+(gx^4+hx^3+ix^2+jx^1+k)(\gamma P +\delta Q)$$
So we will find the formula out of here, or we can find general formula out of here ?
$$(ax^n+bx^n^-^1+cx^n^-^2+dx^n^-^3+ex^n^-^f)(\alpha P+\beta Q)+(gx^n+hx^n^-^1+ix^n^-^2+jx^n^-^3+kx^n^-^w)(\gamma P +\delta Q)$$

 

[tiny-tim]

If there is a formula, then it will be a general formula, yes, and we needn't know the power (strictly, "degree" or "order") of the polynomial (eg. $x^3\,+\,...$, or $x^4\,+\,...$).

 

[Physicsissuef]

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

P.S 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 $$\frac{\beta}{0}=\frac{0}{d}$$

 

[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?