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Question

How are discontinuities created in a rational expression?

 

How are zeros identified in a rational expression?

Answer 1

A "rational expression" consists of a polynomial of a variable (such as x) divided by another polynomial.  So you might have something like this:

 

            x² + 5x + 4

f(x) = ------------------

            x² + 9x + 14

 

Discontinuities occur when the denominator is 0.  That's because you can't divide by 0, so the function isn't defined when the denominator is 0.  There are a number of reasons why a function can be "discontinuous" at a specific value of x, but a common one is when it isn't defined at all for that value of x.

 

The zeros of a rational expression are the same as the zeros of the numerator (since 0 divided by anything is still 0), EXCEPT that you have to exclude any value that is also a zero of the denominator.  That's because you'd have 0/0, and that's not defined either.

 

 

As for what reasons might lead you to want to solve such an equation, I'm not sure I can say with any specificity.  If you're solving a problem and that's the way the math works out, then that's what you've got.  I'm not aware of specific types of problems that lead to a rational expression, but someone else might be.