📚 Topic Summary
Simplifying rational expressions is like reducing fractions in arithmetic, but with polynomials. A rational expression is a fraction where the numerator and denominator are polynomials. To simplify, you factor both the numerator and the denominator and then cancel out any common factors. The key is to factor completely! Remember to state any restrictions on the variable (values that would make the denominator zero).
For example, to simplify $\frac{x^2 - 4}{x^2 + 4x + 4}$, you'd factor the numerator as $(x - 2)(x + 2)$ and the denominator as $(x + 2)(x + 2)$. Then, you can cancel the common factor of $(x + 2)$, leaving you with $\frac{x - 2}{x + 2}$, where $x \neq -2$.
🧮 Part A: Vocabulary
Match each term with its definition:
| Term |
Definition |
| 1. Rational Expression |
A. A value that makes the denominator of a rational expression equal to zero. |
| 2. Factor |
B. An expression in the form $\frac{P}{Q}$, where P and Q are polynomials and $Q \neq 0$. |
| 3. Simplify |
C. To rewrite an expression in its lowest terms. |
| 4. Common Factor |
D. A number or expression that divides evenly into two or more numbers or expressions. |
| 5. Restriction |
E. A factor that appears in both the numerator and the denominator of a rational expression. |
✍️ Part B: Fill in the Blanks
To simplify rational expressions, first, you must _________ both the numerator and the _________. Then, identify and _________ any _________ factors. Remember to state any _________ on the variable.
🤔 Part C: Critical Thinking
Explain in your own words why it is important to identify restrictions when simplifying rational expressions. Give an example to illustrate your point.