To find the domain of a rational expression, you need to figure out which values make the expression undefined. A rational expression looks like this:

$$\frac{P(x)}{Q(x)}$$

Here, ( P(x) ) and ( Q(x) ) are polynomials. The domain includes all real numbers except the ones that make the bottom part (the denominator) ( Q(x) ) equal to zero.

Steps to Find the Domain:

1. Identify the Denominator:
   Look at the bottom part of the rational expression.
   For example, in ( \frac{2x + 3}{x^2 - 4} ), the denominator is ( x^2 - 4 ).

2. Set the Denominator to Zero:
   To see what values to skip, set the denominator equal to zero:

   $$x^2 - 4 = 0$$

   This breaks down to:

   $$(x - 2)(x + 2) = 0$$

   So, ( x = 2 ) and ( x = -2 ).

3. Exclude These Values:
   The values you find (like ( x = 2 ) and ( x = -2 )) tell you what to leave out of the domain. Therefore, for the expression ( \frac{2x + 3}{x^2 - 4} ), we skip ( x = 2 ) and ( x = -2 ).

4. Write the Domain:
   You can express the domain using interval notation. For our example, it looks like this:

   $$(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$$

   This means that any real number is part of the domain except for ( -2 ) and ( 2 ).

Summary:

• The domain of a rational expression includes all real numbers except where the denominator is zero.
• To find the domain, you need to:
  • Identify the denominator.
  • Solve ( Q(x) = 0 ) for the variable.
  • Exclude those values from the domain.

By following these steps, students can effectively find the domain of any rational expression. This is important for doing math correctly and making sure calculations make sense!