Welcome to a new week of class! Tonight, we are moving forward into Chapter 6, covering sections 6-1 through 6-3. These sections lay the groundwork for working with rational expressions, which combines our previous knowledge of factoring with algebraic fractions.
Update on Test Grading
I know many of you are anxious to see your results from the recent test. I am working diligently to finish grading them tonight. However, the process is taking a bit longer than anticipated because I am carefully reviewing the specific steps in your work.
It is very important to me that I don't just mark an answer "right" or "wrong." I am analyzing your logic to figure out exactly what you are doing in your steps. This allows me to award partial credit where you deserve it, ensuring your grade reflects your understanding rather than just the final number.
Chapter 6 Overview: Rational Expressions
Please review the PowerPoint slides linked below for a deep dive into the material. Here are the key concepts we are covering tonight:
Section 6.1: Rational Functions and Simplifying
We begin by defining rational expressions. Remember, a rational expression is essentially a ratio of two polynomials, expressed as:
$$ \frac{P(x)}{Q(x)} \quad \text{where} \quad Q(x) \neq 0 $$
The core skill here is simplifying. To simplify a rational expression, you must factor both the numerator and the denominator completely, and then cancel out common factors. Do not fall into the trap of trying to cancel individual terms that are added or subtracted!
Section 6.2: Multiplication and Division
Once we can simplify, we move to operations. Multiplying rational expressions is straightforward—you multiply across the top and across the bottom, then simplify:
$$ \frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D} $$
Division introduces one extra step. Remember to multiply by the reciprocal of the divisor:
$$ \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} $$
Section 6.3: Addition and Subtraction
This is often the most challenging section because it requires finding a Least Common Denominator (LCD). Just like with numerical fractions, you cannot add algebraic fractions unless the denominators match:
$$ \frac{A}{C} + \frac{B}{C} = \frac{A+B}{C} $$
When denominators are different, you must build equivalent expressions using the LCD before combining the numerators.
Class Resources
Keep up the great work, and keep an eye out for your test grades coming very soon!