Factoring is important in mathematics

[Darkiekurdo]

Hi,
Factoring is important in mathematics so I should know how to factor things. But I don't see how one should factor something! I have looked all over the web but I still don't get it. Could someone show me how factoring works? I would appreciate it.

Thank you.

 

[Hurkyl]

What sorts of things do you want to factor?

 

[Darkiekurdo]

Polynomials.

 

[Hurkyl]

To factor an arbitrary polynomial over a finite field, something like Berlekamp's method is usually used.

To factor arbitrary rational polynomials (or polynomials over a number field), I think the favorite method is to use numerical methods to find a root, and the use lattice methods to recover the minimal polynomial of that root.

 

[Darkiekurdo]

I mostly want to use factorization because I want to find the critical points of a given function so I can use it to find the maxima and minima of the given function.

 

[Hurkyl]

Whoops, I'm wrong about the integer case:
http://en.wikipedia.org/wiki/Polynomial_factorization#Practical_techniques

 

[Darkiekurdo]

To factor an arbitrary polynomial over a finite field, something like Berlekamp's method is usually used.

To factor arbitrary rational polynomials (or polynomials over a number field), I think the favorite method is to use numerical methods to find a root, and the use lattice methods to recover the minimal polynomial of that root.

I'm sorry, but I don't understand this. Could you explain this like you would explain a 14-year old (like me)? I'm sorry for my ignorance.

 

[Hurkyl]

Oh, if you're trying to factor small polynomials by hand, the rational root theorem is one of the most useful techniques.

The thing I mentioned is more of a sledgehammer approach that a computer would use to factor a large polynomial.

 

[matt grime]

I mostly want to use factorization because I want to find the critical points of a given function so I can use it to find the maxima and minima of the given function.

The examples you will do will all be easy to factor by trial and error, or appeal to the quadratic formula. This is because the questions will not be attermpting to find just how good you are at impossible things. You will undoutbedly only have to factor something like x^4+x^2-2, for whcih you will easily recognise 1 and -1 as roots, this allows you to do polynomial division, or less fancily write

(x-1)(x+1)(ax^2+bx+c)=x^4+x^2-2

and multiplying out and equating coefficients shows that a=1, c=2 and you can find b. This means you now have to factor only a quadratic which is easy by anyone's standards since there is a formula for it.