- the top and bottom have only integer coefficients,
- the leading coefficient on the bottom is positive,
- the coefficients have no common integer factor greater than 1, and
- the top and bottom have no common polynomial factor that is not constant.

We add, subtract, multiply, and divide rational expressions using the same techniques as for rational numbers. But systematic factoring is now more important, since less can be done by trial and error. When multiplying or dividing, you should factor all of the numerators and denominators, and there is no need to get common denominators; when adding or subtracting, you only need to factor the denominators at first, and you need to make all of these denominators the same. (You should still factor the numerator of the final answer when adding or subtracting, but that's only to simplify the result.)

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