 We can reduce a fraction to lowest terms, so remember, the fraction a over b is reduced to lowest terms when a and b have no common factor besides 1. Since rational expressions are essentially fractions with polynomials, we'll define the following. The rational expression p of x over q of x is reduced to lowest terms when p of x and q of x have no common factors besides 1. Now, we've reduced a fraction by removing common factors. We made use of the fact that na over nb is the same as a over b. And so we'll produce a rational expression by removing common factors. So, for example, let's try to reduce 12 over 9 minus 3x to lowest terms. Now again, it's helpful to remember that a factor only matters if it's a common factor. So the factors of 12 besides 1 are 2, 3, 4, 6, and 12. And we really only care if one of these is also a factor of 9 minus 3x. And we see that 9 minus 3x is 3 times 3 minus x. So let's rewrite 12 using our factor of 3. So 12 is 3 times 4. Equals means replaceable, so 9 minus 3x can be replaced with 3 times 3 minus x. And now numerator and denominator have a common factor of 3, so we can remove it, leaving. Now a useful thing to remember is that minus a minus b is the same as b minus a. And this gives us a limited ability to rearrange some terms of our expression. And this will be useful on occasion. So let's say we want to simplify 12 minus 3x over x minus 4. So let's factor our numerator. And notice that this factor 4 minus x is almost the same as this factor x minus 4 in the denominator. And remember, if our two factors are the same except for the order of the subtraction, we can reverse the order of subtraction by introducing a factor of minus 1. So negative a minus b is the same as b minus a. And b minus a is the same as negative a minus b. So I can change this 4 minus x into a negative x minus 4. Equals means replaceable, so 12 minus 3x can be replaced with minus 3 times x minus 4. Let's go ahead and reinforce that this x minus 4 in the denominator is a single thing by throwing it inside a set of parentheses. And now the numerator has a factor of x minus 4. The denominator has a factor of x minus 4. And so we can remove the common factors, leaving. And we'll simplify, minus 3 over 1 is the same as minus 3.