[matin_hawk]

I don't know how to do factorisation could somebody help me

[albena]

Hi! In case you mean prime factorisation, take a look at
http://amby.com/educate/math/2-1_fact.html
It contains a good description with examples and check up questions

[CroCivic91]

If you mean like factoring expressions like a(n)*x^n+a(n-1)*x^(n-1)+... +a(1)*x+a(0) = 0 then the answer is a bit trickier. It mainly has to do with dividing polynoms. If you can't divide polynoms, it's going to be real tough to explain.

There is a theorem in elementary mathematics saying something like:
Let f(x) be a polynom (like in the example above). If f(y) = 0, and y is an integer, then y divides a(0).

Now, lets look at the example:
f(x)=x^2+2x+1
By the above theorem, if f(y) = 0, for any y that is an integer, then y has to divide 1 (which is a(0) in our example). Now there are only 2 such numbers, -1 and 1. Lets test it: f(1) = 4, f(-1)=0. Nice, we've found out that f(-1)=0. Now that means that you can divide f(x) by (x-(-1)). Lets divide those 2 polynoms -> (x^2+2x+1)/(x+1) = x+1. Now that means that x^2+2x+1=(x+1)*(x+1)=(x+1)^2.

Next example:
f(x)=x^2+7x+12
By the above theore... such numbers are: 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12. Phew, quite a few of them, eh? Well, quick inspection shows that f(-3)=f(-4)=0, and f(other numbers) is not equal to zero. That means that (x-(-3)) and (x-(-4)) divide f(x). That means that f(x)=(x+3)*(x+4)

Next example:
f(x)=x^4+9*x^3+28*x^2+38x+24
By the above... numbers are: 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 24, -24. Quick check only gives us f(-3)=f(-4)=0. That means that we can divide f(x) by (x+3) and by (x+4). (x+3)*(x+4)=x^2+7x+12, so that means that we can divide f(x) by x^2+7x+12. When we do it - we get f(x)=(x+3)*(x+4)*(x^2+2x+2). Now let's see if we can factor (x^2+2x+2). By the above... numbers are: 1, -1, 2, -2. Quick check gives us that f(1), f(-1), f(2) and f(-2) are all different than zero, meaning that we can't extract another (x-(something)), where that "something" would be an integer.

Well, I know it sounds complicated, but it actually isn't that hard once you learn how to divide polynoms. Also, there are similar rules how to find out if it is possible to extract a factor in the form of (x-p), where p is a fraction. If you're interested in that, let me know, I'll post a follow up on that.

Kresho