Simplifying Rational Expressions
Simplifying a rational expression is done much the same way as simplifying a fraction.
Example 1: The monomials in the numerator and denominator below can be expanded through complete factorization. Then, all of the ones in the form of a number or variable over itself can be divided out. Examples of representations of one include \(\dfrac{3}{3}\) and \(\dfrac{x}{x}\).
\(\dfrac{3x^3\cdot{y^2}}{12x^{4}y}=\dfrac{3\cdot{x}\cdot{x}\cdot{x}\cdot{y}\cdot{y}}{2\cdot{2}\cdot{3}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{y}}=\dfrac{y}{4x}\)
Example 2: This example has polynomials in the numerator and denominator.
First factor, if possible.
\(\dfrac{3x+12}{x^2+7x+12}=\dfrac{3(x+4)}{(x+3)(x+4)}\)
As with the monomials in the first example, simplify by dividing out any forms of one, whether they are monomials or polynomials. In this case, \(\dfrac{x+4}{x+4}\) is a form of 1.
The simplified form of this rational expression is:
\(\dfrac{3}{x+3}\)
Remember only factors can be divided out, not terms by themselves! \(\dfrac{x+4}{x+6}\) The “x’s” cannot be divided out of this expression!
Example 3: This rational expression has polynomials in the numerator and denominator that can be factored. There are multiple forms of one that can be divided out to reveal a much simpler form of the expression.
\(\dfrac{(x^2-4)(x^2-x)}{(x^2-2x)(x^2+4x+4)}=\dfrac{(x+2)(x-2)x(x-1)}{x(x-2)(x+2)(x+2)}=\dfrac{x-1}{x+2}\)
