Topic: Rational Expressions

Subtopic: Operations I - Evaluate & Simplify

**Overview**

Our goal today is to work with algebraic fractions (a.k.a. rational expressions). All the operations you learned to do with numeric fractions in grade school (multiply, "flip and multiply" to divide, LCDs to add, etc.) we will be learning except that the numerator and denominator of our fractions will be polynomials.

Be sure that you recognize __opposites__ (like x-1 and 1-x) and how to handle them when cancelling or finding LCDs.

Don't worry too much about the graphical representation of "excluded values" a.k.a. "domain restrictions", but you should recognize algebraically that there are x-values that can't be plugged into a rational expression because they cause the denominator to be zero (e.g., in the expression 2/(x+5) x cannot be -5) which is undefined. You can find the domain restrictions by factoring the denominator of the rational expression and determining what x-values would make it zero, i.e. set it equal to zero and solve for x.

Caution: when reducing a rational expression like (2x^{2}-50)/(x+5) do not cancel the x's nor reduce the 5's! This is a very common error, tempting, but extremely illegal. Remember that *you can only cancel factors not terms*. So before canceling you must factor completely! Never reduce a rational expression without factoring top and bottom completely first.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 8.1.1 Definition of a rational expression or rational function
- 8.1.2 Graph a rational function electronically
- 8.1.3 Find domain restriction(s) of a rational expression/function algebraically
- 8.1.4 Recognize domain restriction(s) of a rational function by observation of its graph
- 8.1.5 Evaluate a rational expression/function
- 8.1.6 Simplify a rational expression by reducing (cancelling factors)
- 8.1.7 Cancel "opposites" when reducing a rational expression
- 8.1.8 Understand that rational functions can be used to model data

**Terminology**

Define: rational, rational function, domain, domain restrictions (a.k.a. excluded values), vertical asymptote line

**Text Notes**

- Your text may occasionally use the "table" feature of a graphing calculator. You are not required to know how to do this. The analysis the table approach provides can be accomplished algebraically or graphically instead.

- Domain restriction(s) of a rational function is/are represented graphically by a vertical asymptote line at that x-value. This will be covered more thoroughly in a college algebra class. For now it suffices to find the domain restriction(s) algebraically by finding the x-value(s) that make the denominator zero.

- Some texts show graphs that have a vertical asymptote line without actually drawing the line. If you want to do the same thing on your graphing calculator graph in "dot mode" so that the vertical asymptote is not shown as a solid vertical line.