How to Add & Subtract Rational Expressions

In mathematics, the term rational expression denotes a representation of the form p/q where both p and q are polynomials. There are many times when we will need to combine two or more rational expressions through addition or subtraction. The process for doing so is very similar to adding or subtracting standard fractions. Additionally, you will need to utilize the concepts of factoring polynomials, multiplying polynomials, and combining like terms of polynomial expressions.

Things You'll Need

• Paper
• Pencil or Pen

Instructions

1. Write the rational expressions that you wish to combine through addition or subtraction on your paper. To illustrate the technique used to combine two rational expressions, we will use the example 3/(x + 1) – 7/(x + 3).

2. Find the least common denominator for the rational expressions. In our example, that would be (x + 1)(x + 3).

3. Rewrite the rational expressions using the least common denominator. We would obtain 3/(x + 1) – 7/(x + 3) = 3(x + 3)/[(x + 1)(x + 3)] – 7(x + 1)/[(x + 3)(x + 1)] for our example.

4. Combine the rational expressions into one expression now that you have a common denominator. Here, we get 3(x + 3)/(x + 1)(x + 3) – 7(x + 1)/(x + 3)(x + 1) = [3(x + 3) – 7(x + 1)]/[(x + 3)(x + 1)].

5. Simplify your numerator by performing any required multiplication and combining like terms. Our example becomes [3(x + 3) – 7(x + 1)]/[(x + 3)(x + 1)] = (3x + 9 – 7x – 7)/[(x + 3)(x + 1)] = (-4x +2)/[(x + 3)(x + 1)].

6. Factor out any common terms in the numerator. In our example, we could factor out either 2 or -2. We will choose to factor out -2 so that the coefficient of x will be positive. Doing this, we obtain (-4x +2)/[(x + 3)(x + 1)] = -2(2x – 1)/[(x + 3)(x + 1)].

7. Simplify your resulting rational expression if possible. In our example, there is no further simplification that can be done.

Tips & Warnings

• Don’t forget to distribute any negative signs! This is one of the most common mistakes made when combining rational expressions.