Add or subtract, as indicated. See Example 4. (1/(x + z)) + (1/(x - z))

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Identify the given expression: \(\frac{1}{x+z} + \frac{1}{x-z}\).

To add these two fractions, find a common denominator. The denominators are \((x+z)\) and \((x-z)\), so the common denominator is their product: \((x+z)(x-z)\).

Rewrite each fraction with the common denominator: multiply the numerator and denominator of the first fraction by \((x-z)\), and the numerator and denominator of the second fraction by \((x+z)\), giving \(\frac{1 \cdot (x-z)}{(x+z)(x-z)} + \frac{1 \cdot (x+z)}{(x-z)(x+z)}\).

Combine the numerators over the common denominator: \(\frac{(x-z) + (x+z)}{(x+z)(x-z)}\).

Simplify the numerator by combining like terms, then consider if the denominator can be simplified using the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).

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Adding and Subtracting Rational Expressions

To add or subtract rational expressions, they must have a common denominator. This involves finding the least common denominator (LCD) and rewriting each fraction with this denominator before combining the numerators.

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Factoring expressions like x² - z² into (x + z)(x - z) helps identify common denominators and simplifies the process of adding or subtracting fractions. Simplification reduces the expression to its simplest form.

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The difference of squares formula states that a² - b² = (a + b)(a - b). Recognizing this pattern in denominators like x² - z² allows for easier manipulation and finding common denominators in rational expressions.

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