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#1 2017-02-07 13:02:38

Oran2009
Member
Registered: 2016-06-13
Posts: 21

Some Polynomial Problems

1) The polynomial


is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$.

     
     Do I plug in x=4? What would I do???
-------------------------------------------------------------------------------------------------------------------------------------------
2) Suppose the polynomial
is of degree
and satisfies
,
,
, and
.

Determine the value of

.

     
     How to do this?
-------------------------------------------------------------------------------------------------------------------------------------------

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#2 2017-02-07 13:16:01

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Some Polynomial Problems

Hi;


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2017-02-07 13:53:34

Oran2009
Member
Registered: 2016-06-13
Posts: 21

Re: Some Polynomial Problems

I got #1. The answer is just m=5

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#4 2017-02-07 19:18:16

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Some Polynomial Problems

Last edited by thickhead (2017-02-07 19:19:24)


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#5 2017-02-07 23:55:36

iamaditya
Member
From: Planet Mars
Registered: 2016-11-15
Posts: 821

Re: Some Polynomial Problems

bobbym wrote:

Hi;
I am suffering from a migraine so I am a bit weak....

What is migraine?

Last edited by iamaditya (2017-02-07 23:56:12)


Practice makes a man perfect.
There is no substitute to hard work
All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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#6 2017-02-08 00:09:25

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Some Polynomial Problems

Hi;

Without the jargon, it is an intense headache.

The symptoms that one is on the way are bad enough.

http://www.health.com/health/gallery/0, … 78,00.html


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2017-02-08 03:57:54

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Some Polynomial Problems

I feel very sorry for bobbym having my grain.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#8 2017-02-08 11:19:49

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Some Polynomial Problems

Thanks, but it is the consequence of doing all that EM.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2017-02-16 23:20:46

iamaditya
Member
From: Planet Mars
Registered: 2016-11-15
Posts: 821

Re: Some Polynomial Problems

EM...means??


Practice makes a man perfect.
There is no substitute to hard work
All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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#10 2017-02-17 01:11:05

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Some Polynomial Problems

I am glad you asked me that question. EM mean Experimental Mathematics.

I would say the first rule is this:

Get Your Hands Dirty: This is easy and fun to do. Stay loose and experiment. Plug in lots of numbers. Keep playing around until you see a pattern. Then play around some more, and try to figure out why the pattern you see is happening. It is a well-kept secret that much high-level mathematical research is the result of low-tech "plug and chug" methods. The great Gauss, widely regarded as one of the greatest mathematicians in history, was a big fan of this method. In one investigation, he painstakingly computed the number of integer solutions to x^2 + y^2 < 90,000

Today, we can be assisted by a computer to help with this type of research.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2017-03-08 17:45:15

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Some Polynomial Problems

Oran2009 wrote:

1) The polynomial


is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$.

     
     Do I plug in x=4? What would I do???
-------------------------------------------------------------------------------------------------------------------------------------------
2) Suppose the polynomial
is of degree
and satisfies
,
,
, and
.

Determine the value of

.

     
     How to do this?
-------------------------------------------------------------------------------------------------------------------------------------------

Hi Oran2009,
I have come out with an easier solution for (2) in which you don't have to solve simultaneous equations.
Try the polynomial f(x)=A(x-3)(x-4)(x-5)+B(x-3)(x-4)+C(x-3)+2
The remainder 2 for the division by x-3 is already incorporated.
Plug in x=4 you get C.
Plug in x=5you get B.
Plug in x=6 you get A.
then plug in x=0 to get f(0).
Even if it is homework problem lot of time has elapsed and I am giving you only method.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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