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.

$$\dfrac{(x^2-4)(x^2-x)}{(x^2-2x)(x^2+4x+4)}=\dfrac{(x+2)(x-2)x(x-1)}{x(x-2)(x+2)(x+2)}=\dfrac{x-1}{x+2}$$