Welcome back to class! Tonight, we tackled Sections 8.1 through 8.3, moving into the territory of Rational Functions. If the word "rational" reminds you of "ratios" (fractions), you are exactly right. We are essentially dealing with fractions that contain polynomials in both the numerator and the denominator.

Section 8.1: Domain of Rational Functions

A rational function is defined as:

$$f(x) = \frac{P(x)}{Q(x)}$$

Where $P(x)$ and $Q(x)$ are polynomials. The most critical rule to remember here is that division by zero is undefined. Therefore, to find the domain of a rational function, we must set the denominator equal to zero and exclude those values.

For example, if you have $f(x) = \frac{5}{x-3}$, the domain is all real numbers except where $x - 3 = 0$. So, $x \neq 3$.

Section 8.2: Multiplication and Division

When multiplying or dividing rational expressions, the strategy is Factor, Factor, Factor!

  • Multiplication: Factor numerators and denominators completely, then cancel out common factors before multiplying across.
  • Division: Remember to multiply by the reciprocal (Keep-Change-Flip). $$ \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} $$

Section 8.3: Addition and Subtraction

Adding and subtracting is slightly trickier because, just like with regular arithmetic fractions, you need a Least Common Denominator (LCD).

  1. Factor the denominators of all expressions.
  2. Find the LCD.
  3. Rewrite each expression with the LCD.
  4. Combine the numerators (be careful with subtraction signs!).
  5. Simplify the final result if possible.

Homework Assignment

Please practice these concepts with the following problems from your textbook. Consistency is key to mastering factoring and LCDs!

  • Pg. 430-433: #3, 13, 23, 25, 27, 31, 41, 57, 59, 65, 69, 75, 79
  • Pg. 437-439: #5, 19, 23, 37, 51, 55, 57, 63, 67, 71
  • Pg. 448-450: #5, 11, 19, 23, 29, 53, 57, 61, 63, 69, 71

Class Resources

If you missed class or need a refresher on the lecture, please review the notes and videos below:

Don't forget to complete the Weekly Quiz! See you in the next class.