Factor the numerator: \(\frac{(x - 3)(x + 3)}{x - 3}\).

["# Factor the Numerator: Simplifying the Rational Expression (\frac{(x - 3)(x + 3)}{x - 3})", "When working with rational expressions in algebra, one of the most important skills is factoring and simplifying expressions—especially when canceling common factors. A typical expression many students encounter is:", "[\n\frac{(x - 3)(x + 3)}{x - 3}\n]", "While this may seem straightforward, understanding how to properly factor and simplify this expression is crucial for solving equations, analyzing function behavior, and working with rational functions. In this article, we’ll break down how to factor the numerator, simplify the expression, identify restrictions, and explain why cancellation is valid.", "---", "## Step 1: Factor the Numerator", "The numerator is a product of two binomials: ((x - 3)(x + 3)). This is a classic example of a difference of squares, which follows the formula:", "[\na^2 - b^2 = (a - b)(a + b)\n]", "Here, (x) is squared and subtracted from (3^2), so:", "[\n(x - 3)(x + 3) = x^2 - 9\n]", "Thus, the numerator is factored as (x^2 - 9), but understanding its structure as a difference of squares helps recognize opportunities for cancellation.", "---", "## Step 2: Identify Common Factors", "Now the expression becomes:", "[\n\frac{(x - 3)(x + 3)}{x - 3}\n]", "After factoring, we clearly see that ((x - 3)) appears in both the numerator and denominator. This shared factor is the key to simplification.", "Important Note: Cancellation is only valid when the common factor is not zero—that is, when the denominator does not equal zero.", "---", "## Step 3: Simplify by Cancellation (With Restrictions)", "Since ((x - 3)) is common in both numerator and denominator:", "[\n\frac{(x - 3)(x + 3)}{x - 3} = x + 3, \quad \ ext{provided } x <br/>\ne 3\n]", "✅ Simplified Expression: (x + 3)\n⚠️ Restriction: (x <br/>\ne 3) (since division by zero is undefined)", "---", "## Step 4: State the Domain Restriction", "Even though algebraically we cancel ((x - 3)), the original expression is undefined at (x = 3), because it would make the denominator zero. Thus, the simplified function is equivalent to (x + 3) except at (x = 3).", "This highlights a critical concept in rational expressions: simplification changes the domain, though the behavior approximates the original function near the excluded point.", "---", "## Why This Matters", "Factoring and simplifying (\frac{(x - 3)(x + 3)}{x - 3}) isn’t just an academic exercise. It’s essential in:", "- Solving rational equations\n- Analyzing asymptotic behavior and holes in graphs\n- Preparation for calculus operations like differentiation and integration", "Understanding why cancellation works—or when it does not—builds strong algebraic reasoning.", "---", "## Summary", "- The numerator ((x - 3)(x + 3)) factors into a difference of squares: (x^2 - 9), or more meaningfully as a product ((x - 3)(x + 3)).\n- The expression simplifies to (x + 3), but only when (x <br/>\ne 3).\n- Always state domain restrictions after simplifying to preserve mathematical accuracy.\n- Recognizing common factors helps reveal deeper structure and simplifies solving and graphing rational expressions.", "---", "## Key Takeaways for Students", "- Always factor numerators fully before simplifying.\n- Identify common factors before canceling.\n- Never forget to note any values excluded from the domain due to division by zero.\n- Practice with real functions to internalize why simplification has boundaries.", "---", "### Related Keywords for SEO Optimization:", "- Factor numerator\n- Simplify rational expressions\n- Cancel common factors algebra\n- Define domain of rational functions\n- Simplify (\frac{x^2 - 9}{x - 3})\n- Algebraic simplification guide\n- Rational expressions tutorial", "---", "Mastering factoring and simplification loves like this builds a foundation for advanced algebra, calculus, and engineering applications. Start practicing with variations—reduce (\frac{(x - 5)(x + 5)}{x - 5}), (\frac{a^2 - b^2}{a - b}), and more—and watch your confidence grow!"]