Problem 68
Question
Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{2}{x-2}+\frac{3}{x+2}-\frac{x-1}{x^{2}-4}$$
Step-by-Step Solution
Verified Answer
The simplified result is \( \frac{4x - 1}{(x-2)(x+2)} \).
1Step 1: Identify the Expression and Denominator
The given expression is \( \frac{2}{x-2} + \frac{3}{x+2} - \frac{x-1}{x^2-4} \). Notice that \( x^2-4 \) can be factored as \((x-2)(x+2)\). So, the denominators are \(x-2\), \(x+2\), and \((x-2)(x+2)\).
2Step 2: Find a Common Denominator
The least common denominator (LCD) for these fractions is \((x-2)(x+2)\). This means each term must be rewritten with this denominator.
3Step 3: Rewrite Each Fraction with the Common Denominator
- For \( \frac{2}{x-2} \), multiply numerator and denominator by \(x+2\) to get \( \frac{2(x+2)}{(x-2)(x+2)} \).- For \( \frac{3}{x+2} \), multiply numerator and denominator by \(x-2\) to get \( \frac{3(x-2)}{(x-2)(x+2)} \).- The fraction \( \frac{x-1}{x^2-4} \) already has the common denominator.
4Step 4: Combine the Fractions
Now that all fractions share the common denominator, combine them:\[ \frac{2(x+2)}{(x-2)(x+2)} + \frac{3(x-2)}{(x-2)(x+2)} - \frac{x-1}{(x-2)(x+2)} \].
5Step 5: Simplify the Numerator
Simplify the numerators:- \( 2(x+2) = 2x + 4 \).- \( 3(x-2) = 3x - 6 \).Combine them:\( (2x + 4) + (3x - 6) - (x - 1) \) which equals:\( 2x + 4 + 3x - 6 - x + 1 \).Combine like terms: \( (2x + 3x - x) + (4 - 6 + 1) = 4x - 1 \).
6Step 6: Write the Simplified Expression
The final simplified expression is:\[ \frac{4x - 1}{(x-2)(x+2)} \]. This is the simplified form of the original expression.
Key Concepts
Common DenominatorSimplifying FractionsFactoring Polynomials
Common Denominator
When working with rational expressions, the common denominator is essential for performing addition and subtraction. Think of the denominator as the foundation of a fraction—each part of the expression must have a common foundation to be combined properly.
To find a common denominator, identify the individual denominators. Here, they include \(x-2\), \(x+2\), and \((x-2)(x+2)\). The least common denominator (LCD) is simply their product or a factorization that includes all unique factors at the highest power they appear in any of the denominators. In this example, that's \((x-2)(x+2)\).
The concept of common denominator helps you to write all terms over this single denominator, which is necessary for joining them into a single fraction.
To find a common denominator, identify the individual denominators. Here, they include \(x-2\), \(x+2\), and \((x-2)(x+2)\). The least common denominator (LCD) is simply their product or a factorization that includes all unique factors at the highest power they appear in any of the denominators. In this example, that's \((x-2)(x+2)\).
The concept of common denominator helps you to write all terms over this single denominator, which is necessary for joining them into a single fraction.
Simplifying Fractions
Once all fractions share a common denominator, they can be combined by adding or subtracting the numerators. After combining the numerators, the next step is simplifying the resultant fraction. Simplification includes reducing any expressions, combining like terms, and looking for opportunities to minimize the fraction further.
In this exercise, the numerators of the expressions were first expanded and then simplified through addition and subtraction of like terms:
In this exercise, the numerators of the expressions were first expanded and then simplified through addition and subtraction of like terms:
- Expand expressions like \(2(x+2)\) and \(3(x-2)\).
- Combine these with equivalent terms from other fractions.
- Simplify any remaining expressions by combining like terms.
Factoring Polynomials
Factoring polynomials is a crucial step in simplifying rational expressions and finding common denominators. It involves expressing a polynomial as a product of its factors, which are simpler expressions. Often, this makes it easier to identify common terms in rational expressions.
In this case, a noticeable factorization is \(x^2 - 4\), which can be rewritten as \((x-2)(x+2)\). This is a straightforward example of the difference of squares, a common pattern in polynomial factoring. Recognizing these patterns can help you to quickly rewrite expressions in a form that makes them easier to work with.
Factoring helps not only in finding common denominators but also in the step of simplifying the final expression. By factoring, you reveal hidden simplicity that may not have been apparent at first glance. Knowing how and when to factor efficiently is a powerful skill in working with rational expressions.
In this case, a noticeable factorization is \(x^2 - 4\), which can be rewritten as \((x-2)(x+2)\). This is a straightforward example of the difference of squares, a common pattern in polynomial factoring. Recognizing these patterns can help you to quickly rewrite expressions in a form that makes them easier to work with.
Factoring helps not only in finding common denominators but also in the step of simplifying the final expression. By factoring, you reveal hidden simplicity that may not have been apparent at first glance. Knowing how and when to factor efficiently is a powerful skill in working with rational expressions.
Other exercises in this chapter
Problem 68
Solve equation. If a solution is extraneous, so indicate. \(\frac{a-3}{a+1}=\frac{a-6}{a+5}\)
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Simplify each complex fraction. $$ \frac{\frac{7}{a-b}}{\frac{b}{a^{3}-b^{3}}} $$
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Solve each proportion. $$ \frac{-6}{m}=\frac{m}{-6} $$
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Perform each operation and simplify, if possible. See Example 9. $$ \frac{x^{2}-x-12}{x^{2}+x-2} \div \frac{x^{2}-6 x+8}{x^{2}-3 x-10} \cdot \frac{x^{2}-3 x+2}{
View solution