Cancelling Factors

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

$$\frac{\left(x+5\right)\left(x-5\right)}{\left(x+5\right)\left(x-2\right)}$$

x + 5 sits on both, and $\frac{x+5}{x+5}$ is 1 wherever x + 5 is not 0, so it cancels and leaves $\frac{x-5}{x-2}$. Only factors cancel, never terms. In $\frac{x+6}{x+3}$ it is tempting to cross out the x's and get $\frac{6}{3}$, which is 2. But at x = 3 the fraction is $\frac{9}{6}$, 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.