Originally Posted by

**topsquark** Given the stucture of your factors you can work out how to do this step by step. I'm going to start with the original problem and successively replace "+" with "-" in the appropriate places. I'll leave it to you to find the pattern.

The original expression is:

(1+x+x^2+x^3+...)(1+x^3+x^6+x^9+...)(1+x^5+x^10+.. .) = 1 + ...

This will give us a leading term of (+)1, so we want the next term to be negative. The next term will come from the x in the first factor. There are no other terms that produce an x when we multiply this out, so change the sign on the x term in the first factor:

(1-x+x^2+x^3+...)(1+x^3+x^6+x^9+...)(1+x^5+x^10+...) = 1 - x + ...

The only way to produce an x^2 term is the third term in the first factor. We want this to be positive, so leave it alone.

(1-x+x^2+x^3+...)(1+x^3+x^6+x^9+...)(1+x^5+x^10+...) = 1 - x + x^2 + ...

There are two ways to produce the x^3: the fourth term in the first factor and the second term in the second factor. Now, isolate the terms

(1 + x^3 + ...)(1 + x^3 + ...)(1 + x^5 + ...)...

Recall that (1 + x^3)(1 - x^3) = 1 +0*x^3 - x^6

This eliminates the x^3 term. In the same fashion we wish to change one of the two signs in the x^3 terms in the problem. The question is which one? I would suggest that we have a pattern forming in the first factor where each term is "+ - + - ..." so I'm going to change that one.

(1-x+x^2-x^3+...)(1+x^3+x^6+x^9+...)(1+x^5+x^10+...) = 1 - x + x^2 +0*x^3 + ...

You should have the idea by now. If not let me know and I'll generate some more terms.

-Dan