Factoring question - generalized factoring in integers

elegysix

Hello, this is rather complicated to explain so bear with me.

I was wondering about the coefficients of polynomials which are factorable in the integers, meaning polynomials which can be written as (x+a)(x+b) where a and b are integers.

I had a curious idea about letting the x axis represent powers of x, and the y axis representing the value of each coefficient. Looking at a few plots of these coefficients makes me wonder several things. Most importantly though - is it useful to construct an axis based on powers of a variable?

The polynomials are of the form x^2+(a+b)x+ab.
In these graphs, the x axis - 0,1,2 are the powers of x, and y is the value of the coefficient.
Each of these have a value of 1 for the coefficient of x^2, so all lines converge to (2,1).
Each of the graphs has a constant factor a, and b is varied between -5:5, excluding 0.
I've graphed for a=1,2 and 3.
The graphs are basic and do not show which lines are for which polynomials, if I spend more time on it I'll make it do that.

Can I learn anything about factoring in integers from this?

[PLAIN]http://img805.imageshack.us/img805/5228/69023794.jpg [Broken]

[URL=http://imageshack.us/photo/my-images/84/61558448.jpg/]http://img84.imageshack.us/img84/9545/61558448.jpg [Broken]

http://img828.imageshack.us/img828/4904/95017762.jpg [Broken]

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fresh_42

Mentor
2018 Award
Factoring over the integers is the same as factoring over $\mathbb{Q}$ with the usual properties: degree of the polynomial, factoring over $\mathbb{C}$ and see which zeroes are rational, etc.

"Factoring question - generalized factoring in integers"

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