Rational expressions simplifying examples

📚 Quick Study Guide

• 🔍 Rational Expression: A fraction where the numerator and/or denominator are polynomials. Example: $\frac{x^2 + 2x + 1}{x + 1}$
• ➕ Simplifying Strategy: Factor both numerator and denominator, then cancel common factors.
• 💡 Factoring Techniques: Look for common factors, difference of squares ($a^2 - b^2 = (a+b)(a-b)$), perfect square trinomials, and general trinomial factoring.
• 📝 Important Restriction: A rational expression is undefined when the denominator is zero. Always identify any values that make the denominator zero *before* simplifying, as these remain restrictions even after simplification.
• ➗ Example: Simplify $\frac{x^2 - 4}{x^2 + 4x + 4}$. Factor to get $\frac{(x-2)(x+2)}{(x+2)(x+2)}$. Cancel the common factor $(x+2)$ to get $\frac{x-2}{x+2}$, with the restriction that $x \neq -2$.
• ➗ Advanced Tip: Remember to factor out a $-1$ if it helps with canceling like terms. Example: $\frac{a - b}{b - a} = -1$

Practice Quiz

1. Which of the following is a rational expression?
   1. A. $\frac{\sqrt{x} + 1}{x}$
   2. B. $\frac{x^2 + 1}{x - 3}$
   3. C. $\frac{x}{\sqrt{x} - 1}$
   4. D. $\sqrt{\frac{x+1}{x}}$
2. What is the simplified form of $\frac{4x^2y}{6xy^3}$?
   1. A. $\frac{2x}{3y^2}$
   2. B. $\frac{2}{3xy^2}$
   3. C. $\frac{2x}{3y}$
   4. D. $\frac{4x}{6y^2}$
3. Simplify the rational expression: $\frac{x^2 - 9}{x + 3}$
   1. A. $x + 3$
   2. B. $x - 3$
   3. C. $x^2 - 3$
   4. D. $\frac{1}{x+3}$
4. Simplify: $\frac{x^2 + 5x + 6}{x^2 + 4x + 4}$
   1. A. $\frac{x+3}{x+2}$
   2. B. $\frac{x+2}{x+3}$
   3. C. $\frac{5x+6}{4x+4}$
   4. D. $\frac{3}{2}$
5. Simplify the following rational expression: $\frac{2x^2 - 8}{x - 2}$
   1. A. $2x + 4$
   2. B. $2x - 4$
   3. C. $2(x - 2)$
   4. D. $x + 4$
6. Simplify $\frac{x^2 - 25}{x^2 + 10x + 25}$:
   1. A. $\frac{x-5}{x+5}$
   2. B. $\frac{x+5}{x-5}$
   3. C. $\frac{1}{10x}$
   4. D. $\frac{5}{10x}$
7. Simplify: $\frac{3x^2 + 6x}{x^2 - 4}$
   1. A. $\frac{3x}{x-2}$
   2. B. $\frac{3x}{x+2}$
   3. C. $\frac{3}{x-2}$
   4. D. $\frac{3}{x+2}$