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

Verified step by step guidance

1. Identify the given expression: \(\frac{1}{x+z} + \frac{1}{x-z}\).
2. To add these two fractions, find a common denominator. The denominators are \((x+z)\) and \((x-z)\), so the common denominator is their product: \((x+z)(x-z)\).
3. 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 \(\frac{1 \cdot (x-z)}{(x+z)(x-z)} + \frac{1 \cdot (x+z)}{(x-z)(x+z)}\).
4. Combine the numerators over the common denominator: \(\frac{(x-z) + (x+z)}{(x+z)(x-z)}\).
5. Simplify the numerator by combining like terms, then consider if the denominator can be simplified using the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).

Key Concepts

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.

Factoring and Simplifying Algebraic Expressions

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.

Difference of Squares

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.