First, simplify the expression \( R(x) = rac{x^2 - 1}{x + 1} \). Notice that the numerator is a difference of squares:
["# Simplify ( R(x) = \frac{x^2 - 1}{x + 1} ) Using Algebraic Simplification", "When tackling rational expressions in algebra, one of the first and most important steps is to simplify the expression before proceeding with further calculations. In this article, we’ll break down how to simplify\n[\nR(x) = \frac{x^2 - 1}{x + 1}\n]\nby recognizing a key algebraic identity—the difference of squares.", "---", "## Step 1: Recognize the Difference of Squares in the Numerator", "The numerator is ( x^2 - 1 ), which fits the standard difference of squares formula:\n[\na^2 - b^2 = (a - b)(a + b)\n]\nHere, ( x^2 - 1 = x^2 - 1^2 ), so it can be factored as:\n[\nx^2 - 1 = (x - 1)(x + 1)\n]", "---", "## Step 2: Substitute the Factored Form into the Expression", "Now substitute this factorization into the original expression:\n[\nR(x) = \frac{(x - 1)(x + 1)}{x + 1}\n]", "---", "## Step 3: Simplify by Cancelling Common Factors", "The expression now contains ( (x + 1) ) in both the numerator and denominator. As long as ( x + 1 <br/>\neq 0 ) (i.e., ( x <br/>\neq -1 )), we can safely cancel one ( (x + 1) ) term:\n[\nR(x) = x - 1, \quad \ ext{provided } x <br/>\neq -1\n]", "---", "## Step 4: Understand the Domain Restriction", "Although the simplified form is ( R(x) = x - 1 ), it’s critical to note the simplified expression is not defined at ( x = -1 ), since the original denominator becomes zero there. This restriction must be included when interpreting or using the simplified result.", "---", "## Why This Simplification Matters", "Simplifying rational expressions improves clarity and allows easier substitution in functions, limits, and further algebraic manipulations. In this case, recognizing the difference of squares quickly leads to a cleaner and more utility-rich simplified form.", "---", "## Final Simplified Expression", "[\n\boxed{R(x) = x - 1, \quad x <br/>\neq -1}\n]", "---", "## Summary", "- Start with ( R(x) = \frac{x^2 - 1}{x + 1} )\n- Factor numerator using difference of squares: ( x^2 - 1 = (x - 1)(x + 1) )\n- Cancel ( x + 1 ) (with the restriction ( x <br/>\ne -1 ))\n- Final simplified form is ( x - 1 ), excluding ( x = -1 )", "Mastering how to simplify rational expressions like this builds foundational algebra skills essential for calculus and beyond."]
