Adding and Subtracting Rational Expressions

Definition A rational expression is a ratio (fraction) of two polynomials. All rational expressions are valid for all real numbers except those values that make the denominator zero.

are rational expressions whereas are not.

When adding (or subtracting) rational expressions with a common denominator, the process is straightforward: Simply add (or subtract) the numerators, and put the result in a rational expression with the same denominator; i.e.,

where a, b and c are polynomials.

Example 1

Example 2

Note that there is no restriction on x because , as will be shown in another paper.

When performing any operation with rational expressions of a single variable, we must make sure our final answer is simplified:

a. All polynomials are in descending order; i.e., the largest power of the variable appears on the left and as we proceed to the right, the powers of the variable decrease.

b. All leading terms (highest degree terms) in the denominator have positive coefficients.

c. All like (similar) terms have been combined.

d. The greatest common factor of the numerator and denominator is 1; i.e., all common factors have been divided out.

e. Unless told otherwise, the numerator is to multiplied out and simplified, while the denominator may be left in factored form.

The expressions, , are considered to be simplified. The expressions, are not considered to be simplified. The following shows how to convert the latter pair of expressions into simplified forms.

[Factor out -1 from denominator]

[Put in descending order]

[Factor out -1 from (-x + 1)]

[Eliminate the leading neg sign in denom.]

[Simplify]

Example 3

Substract:

Solution: The factorizations of the denominators are

Hence the LCD is:

LCD = (x - 2)(x + 2)(x + 1)

Use these factorizations to convert the rational expressions to rational expressions that have the LCD for their denominators, perform the subtraction, and simplify the result.

[Acceptable as being simplified]

[Preferrable]

Note that the above is valid for all . What this means is that if we substitute any value of

into , and evaluate the expressions, we obtain the same result.

For example, let x = 3. Then we obtain

and