Detailed examples for adding rational expressions with shared denominators

📚 Quick Study Guide

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• Adding Rational Expressions: When adding rational expressions with shared denominators, you simply add the numerators and keep the common denominator.
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• General Formula: $\frac{A}{C} + \frac{B}{C} = \frac{A+B}{C}$, where A, B, and C are polynomials, and C ≠ 0.
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• Simplifying: After adding, always simplify the resulting rational expression if possible by factoring and canceling common factors.
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• Common Denominator: Ensure that the denominators are exactly the same before adding the numerators.

Practice Quiz

1. Question 1: What is the result of $\frac{3x}{x+2} + \frac{5x}{x+2}$?
• A: $\frac{8x}{x+2}$
• B: $\frac{8x^2}{x+2}$
• C: $\frac{8x}{2x+4}$
• D: $\frac{15x^2}{(x+2)^2}$
2. Question 2: Simplify $\frac{x+3}{x-1} + \frac{2x-1}{x-1}$.
• A: $\frac{3x+2}{x-1}$
• B: $\frac{3x+4}{x-1}$
• C: $\frac{3x+2}{2x-2}$
• D: $\frac{x+2}{x-1}$
3. Question 3: Evaluate $\frac{4}{x+1} + \frac{x}{x+1}$.
• A: $\frac{4x}{x+1}$
• B: $\frac{4+x}{x+1}$
• C: $\frac{4x}{2x+2}$
• D: $\frac{4}{x+1}$
4. Question 4: Combine $\frac{x^2}{x+5} + \frac{25}{x+5}$.
• A: $\frac{x^2+25}{x+5}$
• B: $x-5$
• C: $\frac{(x+5)^2}{x+5}$
• D: All of the above
5. Question 5: What does $\frac{7x-2}{x+3} + \frac{-2x+1}{x+3}$ equal?
• A: $\frac{5x-1}{x+3}$
• B: $\frac{9x-3}{x+3}$
• C: $\frac{5x-1}{2x+6}$
• D: $\frac{5x+1}{x+3}$
6. Question 6: Find the sum: $\frac{x^2 + 1}{x-2} + \frac{-5}{x-2}$.
• A: $\frac{x^2 - 4}{x-2}$
• B: $x+2$
• C: $\frac{x^2 + 6}{x-2}$
• D: A and B
7. Question 7: Simplify $\frac{x}{x^2-1} + \frac{1}{x^2-1}$.
• A: $\frac{1}{x-1}$
• B: $\frac{x+1}{x^2-1}$
• C: $\frac{1}{x+1}$
• D: A and B