Factoring question - generalized factoring in integers

In summary, the conversation discusses the concept of factoring polynomials over the integers using graphs to represent the coefficients. The graph shows that the coefficients of x^2 are always 1 and the values for the remaining coefficients vary based on the values of a and b. The conversation also raises questions about the usefulness of constructing an axis based on powers of a variable and whether this approach can provide insights into factoring over the integers.
  • #1
elegysix
406
15
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.

Does anyone know anything about or like this? any comments?
Can I learn anything about factoring in integers from this?


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

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


http://img828.imageshack.us/img828/4904/95017762.jpg
 
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  • #2
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.
 

1. What is factoring in integers?

Factoring in integers is the process of breaking down a number into its prime factors. This means finding the prime numbers that can be multiplied together to get the original number.

2. Why is factoring important?

Factoring is important in many areas of mathematics, including number theory, cryptography, and algebra. It can also be used to simplify complicated expressions and solve equations.

3. What is generalized factoring?

Generalized factoring is the process of finding the factors of a polynomial with more than two terms. This involves factoring out a common factor or using techniques such as grouping or the quadratic formula.

4. How do you factor a polynomial?

To factor a polynomial, you need to look for any common factors and use techniques such as grouping or the quadratic formula to find the remaining factors. It may also be helpful to use a factoring calculator or solve the polynomial by trial and error.

5. What is the difference between factoring and simplifying?

Factoring is the process of breaking down a number or expression into its factors, while simplifying involves reducing an expression to its simplest form. Factoring is a specific type of simplification that involves finding the prime factors of a number or expression.

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