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#1 2012-09-24 20:15:19

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,572
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The three zeros of a polynomial

The two zeros of a polynomial are

and

and the polynomial is

What is the third zero?

My approach to such kinds of problem is to divide the polynomial by the two given factors to obtain the other factor and then the zero.

Can Someone suggest a quicker and more efficient method?

Last edited by Agnishom (2012-09-24 20:16:07)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#2 2012-09-24 21:43:15

Au101
Member
Registered: 2010-12-01
Posts: 286

Re: The three zeros of a polynomial

I'm not sure what exactly your approach would be. Originally, I just posted my approach:

If 2 is a zero of the polynomial, then (x - 2) is a factor.

(By polynomial long-division)

Therefore, the other zero is -1

But having re-read your first post, I think that might be what you would have done anyway. I'm not sure that I know any more efficient method. But I suggest you just divide once and then factorise, that - at least - might make things a little faster?

Last edited by Au101 (2012-09-24 21:46:43)

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#3 2012-09-24 22:01:08

bob bundy
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Registered: 2010-06-20
Posts: 6,374

Re: The three zeros of a polynomial

hi Agnishom and Au101

That approach looks like the most straight forward to me.

For this particular problem

you might notice that

which means you know (x+1) is factor straight away by the factor theorem.

In general, dividing by known factors is the way.

Bob

ps.  For typical exam questions, they cannot choose factors that would take a long time to find, so I always do a quick mental check for x = +/-1, +/-2, +/-3.  If I haven't found a factor by then  I do another quicker question first.


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

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#4 2012-09-24 22:20:17

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,572
Website

Re: The three zeros of a polynomial

So, Is division the only way?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#5 2012-09-24 23:18:48

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,374

Re: The three zeros of a polynomial

Well, as you know that cubic = linear x quadratic you can sort of figure out the quadratic coefficients as you go.  It takes less time but amounts to short cut division so it's not really a new method. 

Also you could call the quadratic coefficients a, b and c and do a bit of algebra to get them, but it's still pretty much the same.

In short, I think you have the optimum method already.

Bob


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

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#6 2012-09-25 00:06:12

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: The three zeros of a polynomial

Hi;

You can use the theory of equations to try another way. It says there is a relationship between the roots and the coefficients.

The sum of the roots of a nth degree polynomial

are equal to

So you have this equation to solve

solving for r3 you get r3 = -1 which is the third root.


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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#7 2012-09-26 10:11:25

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: The three zeros of a polynomial

Hi bobbym!

I've never seen the generalized "sum of roots" formula you gave in post#6.  That's really nifty!
Thanks for sharing it.  smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#8 2012-09-26 11:23:16

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: The three zeros of a polynomial

Hi noelevans;

It is nifty. Neat and clean and simple enough for everyone to grasp. That is what math should be like.


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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#9 2012-09-26 14:48:05

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: The three zeros of a polynomial

Hi again bobbym!

I couldn't agree more that math should be neat, clean and simple enough for everyone to grasp.
That's been my passion and my focus most of my career --- viewing math as a language and
trying to see if the symbolism, definitions, algorithms, etc. can be simplified and improved.  It
really needs close scrutiny since it evolved over many centuries by folks that had no chance to
communicate with each other to try to make it really consistent, coordinated, correct, concise, and
any other word we can think of that starts with a "c".

wave


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#10 2012-09-26 20:12:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: The three zeros of a polynomial

Hi;

Clever and compact.


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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