Topic: Rational Expressions
Subtopic: Operations III - Add & Subtract
Overview
In this lesson we add and subtract rational expressions including fractions that have a common denominator and those that don't. When adding rational expressions that have the same denominator, simply add the numerators, simplify, and that result will be over the common denominator. When subtracting be careful to distribute the minus sign through the entire numerator of the fraction it precedes.
Adding/subtracting rational expressions that have different denominators is more complicated. Recall that adding numeric fractions that have different denominators like 5/6 + 2/15 requires finding an LCD, building each fraction up to have that LCD, adding the fractions, and simplifying the answer. Similarly for rational expressions we must do all these steps except that the numerators and denominators will be algebraic expressions and the LCD is likely to be a factored polynomial.
To add/subtract rational expressions that have different denominators:
- Factor all denominators completely.
- Find the LCD. Watch for opposites!
- Build each fraction up to have the LCD (by multiplying by 1).
- Combine the fractions together into one (watch the signs!).
- Simplify the numerator completely.
- Reduce the fraction by factoring and canceling if possible.
Objectives
By the end of this topic you should know and be prepared to be tested on:
- 8.3.1 Add or subtract rational expressions that have the same denominator
- 8.3.2 Add or subtract rational expressions that have different denominators using their LCD as needed
- 8.3.3 Deal with opposites when finding the LCD of rational expressions being added or subtracted
Terminology
Define: LCD = lowest common denominator, "build up" a fraction to have a given denominator
Text Notes
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 7.3-7.4
- Pay special attention to the examples that contain "opposites". Watch your signs! There is lots of factoring throughout these sections. Factor carefully!
- ch 7.3 pg 482 cautions you against a common sign error when subtracting. Don't forget to distribute the minus sign throughout the numerator when combining the fractions.
- ch 7.4 is a very important section. You must learn to find the LCD, build each fraction up to have the LCD by multiplying each fraction by an expression equivalent to 1 e.g. (x-3)/(x-3), combine into a single fraction, simplify top and bottom, factor each completely, and reduce if necessary. These problems can get quite long! Plan your schedule accordingly.