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.
Writing a Rational Expression in Lowest Terms
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:
6x2 -21x - 12 = 3(2x2 - 7x - 4) = 3(x - 4)(2x + 1).
Factor the denominator:
54x2 +45x + 9 = 9(6x2 + 5x + 1) = 9(3x + 1)(2x + 1).
Cancel out common factors:
= 
.
Example 2: Write
in lowest
terms.
Factor the numerator:
x3 - x = x(x2 - 1) = x(x + 1)(x - 1).
Factor the denominator:
6x4 +2x3 -8x2 = 2x2(3x2 + x - 4) = 2x2(x - 1)(3x + 4).
Cancel out common factors:
= 
.
