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 (2x2-50)/(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
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 7.1
- pg 463 (and throughout the text) Using the "table" feature of your graphing calculator is completely optional. The analysis the table approach provides can be accomplished algebraically or graphically instead.
- pg 464 Notice that the 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.
- pg 464 The graph of this function is drawn in "dot mode" so that the vertical asymptote is not shown as a vertical line. If you recreate this graph on a calculator you are likely not in "dot mode" and the domain restriction will be represented as a vertical line at x=4. Note that this vertical line represents the domain restriction and is not part of the actual graph of the function. Note also that this x-value can be found by setting the denominator equal to zero and solving: 7x-28=0 --> 7x=28 --> x=4 is the domain restriction and equation of the vertical asymptote line.
- pg 467 has a very important "caution"! This is a very common error. Remember, you can only cancel factors not terms.
- example 5 I agree with the author that the easiest way to reduce the opposites (like x-3 and 3-x) is to cancel them leaving a -1 factor on the top or bottom. However, if you are used to factoring a -1 out from the top or bottom, or if you are used to multiplying the fraction by -1/-1, then please stick to the method that works best for you.