When we discuss a rational expression in this chapter, we are referring to an expression whose numerator and denominator are (or can be written as) polynomials. For example, and are rational expressions.

To write a rational expression in lowest terms, we must first find all common factors (constants, variables, or polynomials) or the numerator and the denominator. Thus, we must factor the numerator and the denominator. Once the numerator and the denominator have been factored, cross out any common factors.

*Example 1*: Write
in
lowest terms.

Factor the numerator:
6*x*
^{2} -21*x* - 12 = 3(2*x*
^{2} - 7*x* - 4) = 3(*x* - 4)(2*x* + 1)
.

Factor the denominator:
54*x*
^{2} +45*x* + 9 = 9(6*x*
^{2} + 5*x* + 1) = 9(3*x* + 1)(2*x* + 1)
.

Cancel out common factors:
=
.

*Example 2*: Write
in lowest
terms.

Factor the numerator:
*x*
^{3} - *x* = *x*(*x*
^{2} - 1) = *x*(*x* + 1)(*x* - 1)
.

Factor the denominator:
6*x*
^{4} +2*x*
^{3} -8*x*
^{2} = 2*x*
^{2}(3*x*
^{2} + *x* - 4) = 2*x*
^{2}(*x* - 1)(3*x* + 4)
.

Cancel out common factors:
=
.