Math Question

[rjedi]

Yes I could try to find a math forums somewhere but I sure someone might be able to help me and I already feel comferatable posting in this forums so...

I am in Pre-calculus and we are studying finding zeros of polynomials. My teacher is teaching us just the way to find all the possible zeros with the factors of the leading coefficent and the factor of the constant.

Well i decided that was a waste of time going through each one when we have close to 32 factors for each one and found out about the Descarte's rule of signs and the lower and upper bound

my question is, Is there an equation that I can put the equation in question in or variables from it in and figure out the zeros that way? Or am I better off using the Descarte's rule of signs and just having to do 4 or 5 factors instead of 32 or 50?

[Krelar]

my question is, Is there an equation that I can put the equation in question in or variables from it in and figure out the zeros that way? Or am I better off using the Descarte's rule of signs and just having to do 4 or 5 factors instead of 32 or 50?

There is no general formula you can use.

I would assume you already know the quadratic formula for solving 2nd degree polynomials.

There is a cubic formula for 3rd degree polynomials
http://en.wikipedia.org/wiki/Cubic_f...rmula_of_roots

As you can see just jumping from 2nd to 3rd degree is really messy.

On a side note if your pre-calculus curriculum is anything like others I've seen you will probably be learning Descarte's rule in class next/soon.

[SisAmethyst]

Descartes Rule is a technique for determining the number of positive or negative real roots of a polynomial. In other words it not tell you where they are but a prediction how many are there.

see http://en.wikipedia.org/wiki/Descartes%27_rule_of_signs

What you probably want is next to Factoring something like Polynomial Division

see http://en.wikipedia.org/wiki/Division_polynomials

But I guess it is more important to understand what your teacher want you to learn instead of just trying to use an easy key to open a door (like the Gauss summation of 1+2+3..+n)

The Factoring you mentioned is the point that the only way a product can become 0 is if at least one of the factors are 0. Which in other words mean the only way the polynomial function can become 0 is if at least one of its factors are 0.

Polynom : x² + x - 20
Factors : (x - 4)(x + 5) = 0
Polynom get 0 at : x = -5 and x = +4

The tricky part here is to get the Factors right, but the final Factors are pretty easy to handle.

[Janth]

The easiest way to find zeros is with technology. A TI-83 can help.

[jwdaniels]

When trying to find zeroes, I suggest you start between -1 and 1 and then check every 10th digit thereafter in either the positive or negative direction. You will soon have all the zeroes you need.

[Rawel_San]

Where I teach a TI-83 is not an option (I think all college math courses forbid the use
of calculators, I mean let's face it my mobile phone could solve all the problems in
the standard calculus book).

Using descarte's rule in my opinion actually is quite slower
then guessing the roots from the factors. Don't forget that at least some of the roots have to be small (since the constant term is the product of the roots so at least one of
the roots has size at most n-th root of the constant term where n is the degree of the
polynomial).

Once you get one root you can use polynomial division to lower the degree
of the polynomial.
Just my 2 cents,
Rawel

[rjedi]

We are not allowed to use calculators unless specifically stated and my teacher upon asking about an easier way pointed me more or less to the rule of signs and told me she does not teach that.. and thanks for the cubic furmula I will try that out

Amethyst I know its used for guessing the number of factors but with the lower and upper bounds could I not effectivily split the work in more then half?
Say I have the possible factors (+ or -) 1 - 50 and I plug in 25 in synthetic division and find it is a lower bound the plug in 40 and so on until I find the positive zero(s) then do the same thing with the negative zero(s)

So instead of working out 100 synthetic division problems I can work out 5 or 6



And I understand the p/q and guess and check but we have been doing that for a week now and I hate working out so many problems

[Quikster]

Used a TI83 all through cal I and chem I & II. Sometimes its much easier to work on paper though. I needed a totally separate degree to use the darn calculator.

[SisAmethyst]

hehe I still have my FX-5500 from my University time and yes Quikster, for some of the functions you really need a separate course to know how to use it

@rjedi ... the problem is that as Descartes Rule only predict the boundaries of how often the curve crosses the axis. It somehow limit your effort as you not need to look for values out of its scope. That mean if you know that there are a maximum of three you not need to look for a fourth. Where and which value is still not clear by this, but in school most of the values as already mentioned are most of the time anyway in a rather small area to keep the numbers of the multiplication in a reasonable range.

Its already 4 o'clock in the night over here so I have mercy bothering my brain too much

I guess your idea to split the work in half is somewhere based on the game 'guess a number'. Something like pick a number between 1..100 and I tell you if it is upper or lower than that till you found the number. You will then probably pick 50. However in the worst case scenario it is either 1 or 100 which mean you still have to go through a couple of trials. On the other side with each found factor it may get easier.

The polynomial division let you then for example reduce the complexity but for that you usually need at least one factor.

Polynomial : x³ - 4x
Reduced : x² - 4

Well, for the above example the reduction is not really needed but x = 0 is a valid Factor.

The problem is a mathematical curve is rarely as clean as a sinus wave. Which mean you could have all the points of the zeros close to each other or spread over the whole length. As the Descartes Rule only say that there are n numbers of points on this axis but not where they are you still have to find the location.

One way I go is that very often the number itself is hidden in the formula. Like in my first example of the other post the 20 already ask for values like 4 and 5 to try, or in the one above the 0, -2 and +2.

One more complex example of the Descartes rule:

f(x) = 5x^7 + 6x^6 - 9x^4 - 4x^3 - 11x^2 + 7x - 3

were Descaret say it may have either 3 or 1 positive and 4, 2 or 0 negative. Well, that would mean in the first place you would still need to guess which are the 7 points of zero when in fact it only has 1 single positive. But even with the information that it only has a single positive you still not know where exactly on the graph this happens.

[Wrendd]

Math? ummmm--uhhhhhh..yeah, mathz... let me help with thaaaa....*head aspoldes*

Sorry about that. Beyond figuring out drug dosages and dopamne drips my math skills have atrophied beyond hope of salvation. Luckily there are many people here that can and will help you. Good luck