Cancelling Factors

A rational expression is a fraction with polynomials on top and bottom, like (x^2 - 25)/(x^2 + 3x - 10). You simplify it the way you simplify 6/9: find a factor the top and bottom share, and cancel it. Factor both first. The top is a difference of squares, (x + 5)(x - 5), and the bottom needs two numbers that multiply to -10 and add to 3, so it is (x + 5)(x - 2).

((x + 5)(x - 5))/((x + 5)(x - 2))

x + 5 sits on both, and (x + 5)/(x + 5) is 1 wherever x + 5 is not 0, so it cancels and leaves (x - 5)/(x - 2). Only factors cancel, never terms. In (x + 6)/(x + 3) it is tempting to cross out the x's and get 6/3, which is 2. But at x = 3 the fraction is 9/6, which is 3/2. The x there is added, not multiplied, so it is not a factor of the whole top or the whole bottom.

That is why factoring comes first: it shows what multiplies everything. If no factor is shared, the fraction is already as simple as it gets.