simplifying rational expressions
Simplifying a rational expression means cleaning it up by cancelling whatever the top and bottom have in common — exactly the way 6/8 becomes 3/4 once you notice both share a factor of 2. The catch is that with polynomials you cannot cancel terms; you can only cancel factors.
The method is: factor the numerator completely, factor the denominator completely, then strike out any factor that appears in both. What remains is the simplified form. For instance (x^2 - 4)/(x^2 - 4x + 4) factors to (x - 2)(x + 2) / (x - 2)(x - 2), and the shared (x - 2) cancels to leave (x + 2)/(x - 2).
One honest warning: cancelling can quietly erase information. In the example above, the original expression was undefined at x = 2 (the bottom was zero there), and the simplified version still is — but you must keep that restriction in mind, because the cancelled factor no longer announces it. Simplified and original agree everywhere they are both defined, not at the lost point.
(3x + 6)/(x^2 - 4) = 3(x + 2) / [(x - 2)(x + 2)] = 3/(x - 2), valid for x ≠ -2 and x ≠ 2.
Factor first, then cancel the shared (x + 2).
The classic mistake is cancelling across a plus sign: in (x + 2)/x you cannot cancel the x's, because x is a term, not a factor. Only factors — things being multiplied — may be cancelled.