# Factoring question - generalized factoring in integers

• elegysix
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.
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.

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

Last edited by a moderator:
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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