Adding & Subtracting Rational Expressions: Worksheet & Answers
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Adding and subtracting rational expressions can often feel daunting, but with the right approach, it becomes a lot more manageable! In this article, we'll guide you through the processes involved in adding and subtracting rational expressions, provide helpful worksheets, and present the answers for you to check your work. Let’s dive into the world of rational expressions! 📚✨
What Are Rational Expressions?
Rational expressions are fractions where the numerator and the denominator are both polynomials. They can be as simple as ( \frac{3}{x} ) or more complex like ( \frac{x^2 + 3x + 2}{x^2 - 1} ). The key here is that these expressions can be manipulated just like numerical fractions.
Adding Rational Expressions
To add rational expressions, follow these steps:
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Find a Common Denominator: Just like with numerical fractions, you need a common denominator to add rational expressions. This often means factoring the denominators to identify their least common multiple (LCM).
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Rewrite Each Expression: Adjust each expression so that they both have the common denominator.
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Combine the Numerators: Once you have a common denominator, add the numerators while keeping the denominator the same.
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Simplify: Finally, simplify the resulting expression if possible.
Example:
Let’s add the following expressions: ( \frac{2}{x + 2} + \frac{3}{x} )
- Common Denominator: The common denominator will be ( x(x + 2) ).
- Rewrite:
- ( \frac{2}{x + 2} = \frac{2x}{x(x + 2)} )
- ( \frac{3}{x} = \frac{3(x + 2)}{x(x + 2)} = \frac{3x + 6}{x(x + 2)} )
- Combine: [ \frac{2x + (3x + 6)}{x(x + 2)} = \frac{5x + 6}{x(x + 2)} ]
- Simplify: In this case, the expression is already simplified.
Subtracting Rational Expressions
Subtracting rational expressions follows the same steps as adding, but with a slight adjustment in the final step.
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Find a Common Denominator: This is the same as in addition.
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Rewrite Each Expression: Rewrite each rational expression with the common denominator.
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Subtract the Numerators: Subtract the numerators this time.
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Simplify: Simplify the resulting expression if needed.
Example:
Let’s subtract the following expressions: ( \frac{5}{x + 3} - \frac{2}{x} )
- Common Denominator: The common denominator will be ( x(x + 3) ).
- Rewrite:
- ( \frac{5}{x + 3} = \frac{5x}{x(x + 3)} )
- ( \frac{2}{x} = \frac{2(x + 3)}{x(x + 3)} = \frac{2x + 6}{x(x + 3)} )
- Subtract: [ \frac{5x - (2x + 6)}{x(x + 3)} = \frac{5x - 2x - 6}{x(x + 3)} = \frac{3x - 6}{x(x + 3)} ]
- Simplify: This can be simplified further: [ \frac{3(x - 2)}{x(x + 3)} ]
Practice Worksheet
Now it’s time to practice what you’ve learned! Here’s a worksheet with a few problems for you to solve. Try to add or subtract the given rational expressions.
| Problem Number | Problem |
|---|---|
| 1 | ( \frac{4}{x} + \frac{5}{x^2} ) |
| 2 | ( \frac{7}{x - 1} - \frac{3}{x + 1} ) |
| 3 | ( \frac{2}{x + 4} + \frac{1}{x - 4} ) |
| 4 | ( \frac{6}{x^2 + 3x} - \frac{4}{x^2} ) |
| 5 | ( \frac{5}{x - 2} + \frac{3}{x + 2} ) |
Answers to the Worksheet
Ready to check your answers? Here’s how the problems from the worksheet turn out.
| Problem Number | Answer |
|---|---|
| 1 | ( \frac{4x + 5}{x^2} ) |
| 2 | ( \frac{7(x + 1) - 3(x - 1)}{(x - 1)(x + 1)} = \frac{10}{(x - 1)(x + 1)} ) |
| 3 | ( \frac{2(x - 4) + (x + 4)}{(x + 4)(x - 4)} = \frac{3x - 4}{x^2 - 16} ) |
| 4 | ( \frac{6x - 4(x + 3)}{x^2(x + 3)} = \frac{-4}{x^2(x + 3)} ) |
| 5 | ( \frac{5(x + 2) + 3(x - 2)}{(x - 2)(x + 2)} = \frac{8x - 4}{x^2 - 4} ) |
Important Notes
- Factoring: Always factor your denominators as much as possible when looking for a common denominator. It makes the process smoother!
- Undefined Values: Remember that rational expressions are undefined for values that make the denominator zero. Be sure to state these restrictions whenever necessary.
- Practice: The more you practice, the easier it will become to add and subtract rational expressions!
With this detailed guide and worksheets, you can tackle adding and subtracting rational expressions with confidence! Happy studying! 📝💡