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#1 2014-02-02 07:28:49

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Polynomial Ecuation

Hello guys long time no see,got a new exercise:

On R[X] f=X^4-aX^3-aX+1

Proove that if |a|<1 then |x1|=|x2|=|x3|=|x4|=1;
Good luck.

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#2 2014-02-03 07:33:29

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Re: Polynomial Ecuation

Ok guys a second problem

f(1)=8
f(2)=-5
f(3)=0
f(4)=35

f=?

The thing is i know how to solve the problem but i cannot really solve the system.
f(3)=0 -> x1=3 solution ->f = (X-3)(aX^2+bX+c)==>
==>f=aX^3+(b-3a)X^2+(c-3b)X-3c
so f(3)=0 ==>-3c=0 =>c=0

f(1)=8 ==> a+b=-4
f(2)=-5 ==>-4a-2b=-5==>4a+2b=5
f(4)=35 ==> 16a+4b=35

Can anyone solve the system or at least tell me where i'm wrong ? I'm getting very mad here.

Last edited by Yusuke00 (2014-02-03 09:56:58)

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#3 2014-02-03 08:46:17

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,569

Re: Polynomial Ecuation

Hi;

f(1)=9
f(2)=-5
f(3)=0
f(4)=35


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#4 2014-02-03 09:57:39

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Re: Polynomial Ecuation

bobbym wrote:

Hi;

f(1)=9
f(2)=-5
f(3)=0
f(4)=35

Could you describe your solution step by step please? For f(1)=8 not 9 sorry.

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#5 2014-02-03 14:08:06

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,569

Re: Polynomial Ecuation

Hi;

After substitution into the general form you get.

Can you solve that or do you need a little more?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#6 2014-02-03 19:03:36

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Re: Polynomial Ecuation

Ok i got those ecuations either.W/E i'm stupid.Thank you.

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#7 2014-02-03 20:31:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,569

Re: Polynomial Ecuation

Are you okay with getting the solutions? if you need help just post back.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#8 2014-02-04 02:03:54

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Re: Polynomial Ecuation

Yes i had the same ecuations,but i've been tired and just couldn't focus yesterday.Got them now thank you.

By the way,any clues on 1st problem?

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#9 2014-02-04 02:08:53

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,390

Re: Polynomial Ecuation

hi Yusuke00

I didn't post earlier because I do not understand what you mean by X1 X2 etc.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#10 2014-02-04 04:33:44

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Re: Polynomial Ecuation

x1 are the solutions to the ecuation like...
f(x1)=0,f(x2)=0,f(x3)=0,f(x4)=0;
Sorry that's how i know to note them.

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#11 2014-02-04 04:57:52

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,390

Re: Polynomial Ecuation

hi Yusuke00

That's what I thought you meant and that's where my difficulty began.

If you go to this page of the MIF site

http://www.mathsisfun.com/data/function … x)&func2=2

you will find a very useful function grapher.  It even allows you to put in an 'a' and adjust its value using the slider.

This shows no roots at all.

Bob

View Image: yusuke.gif

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#12 2014-02-04 05:19:46

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Re: Polynomial Ecuation

For a>=1 there are real solutions.For a<1 there are not.That function grapher is for R not for C.

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#13 2014-02-04 05:48:53

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Re: Polynomial Ecuation

But it's very helpful,i like it.Thank you very much ^_^.

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#14 2014-02-04 06:47:36

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,390

Re: Polynomial Ecuation

On R[X] f=X^4-aX^3-aX+1

So what does R[X] mean ?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#15 2014-02-04 07:53:08

Yusuke00
Member
Registered: 2013-11-19
Posts: 43

Re: Polynomial Ecuation

It means that f(x) =y,y is part of R.

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