\[ rac{(x - 3)(x + 3)}{x - 3} \]

["# Simplify and Understand the Expression (\frac{(x - 3)(x + 3)}{x - 3})", "When studying algebra, one common expression that students encounter is:", "[\n\frac{(x - 3)(x + 3)}{x - 3}\n]", "At first glance, it appears to be a complex fraction involving multiplication in the numerator and a linear term in the denominator. However, simplifying this expression unlocks deeper mathematical understanding and is essential for solving equations, working with rational functions, and identifying undefined behavior.", "## Simplifying the Rational Expression", "The key to simplifying this fraction lies in recognizing common factors in the numerator and denominator.", "### Step 1: Factor the numerator", "The numerator is ((x - 3)(x + 3)), which is a product of two binomials. Crucially, this matches a factor in the denominator: (x - 3).", "Thus, the expression becomes:", "[\n\frac{(x - 3)(x + 3)}{x - 3}\n]", "### Step 2: Cancel common factors", "Since (x - 3) appears in both the numerator and the denominator (and assuming (x <br/>\neq 3) to avoid division by zero), we can cancel one instance of (x - 3), provided (x <br/>\ne 3):", "[\n\frac{(x - 3)(x + 3)}{x - 3} = x + 3 \quad \ ext{for} \quad x <br/>\ne 3\n]", "### Step 3: Domain restriction", "Even though the expression simplifies to (x + 3), it’s critical to note the restriction (x <br/>\ne 3), because when (x = 3), the original denominator becomes zero, making the expression undefined.", "---", "## Understanding Interpretation and Use", "### 1. Simplification in Algebraic Manipulation", "This expression demonstrates how cancellation rules apply to rational functions. Your algebra skills will strengthen by recognizing like factors and safely simplifying complex fractions while remembering domain restrictions.", "### 2. Graphical Representation", "The function (f(x) = \frac{(x - 3)(x + 3)}{x - 3}) is equivalent to the linear function (f(x) = x + 3), except at (x = 3), where there is a hole (removable discontinuity) rather than a vertical asymptote. Plotting both the original and simplified forms reveals this discontinuity clearly.", "### 3. Importance in Equation Solving", "Equations involving such expressions often reduce cleanly upon simplification. However, always identify and exclude values that make the original denominator zero. This practice prevents extraneous solutions and strengthens problem-solving rigor.", "---", "## Final Simplified Form", "[\n\boxed{ \frac{(x - 3)(x + 3)}{x - 3} = x + 3 \quad \ ext{when} \quad x <br/>\ne 3 }\n]", "This simplified expression is simpler and more intuitive but always remember: the domain excludes (x = 3). Understanding simplification in rational expressions is foundational for calculus, algebra progressions, and real-world modeling.", "---", "### Want to practice more?", "Try evaluating the original expression at several values (e.g., (x = 0), (x = 2), (x = 3), and (x = 4)) and verify how simplification behaves—especially noting the undefined point at (x = 3).", "---", "Keywords: rational expression simplification, solve (x - 3)(x + 3)/(x - 3), domain restrictions, cancellation of factors, algebraic simplification, function analysis, x ≠ 3, hole in graph, simplified rational function", "Meta Description:\nLearn how to simplify (\frac{(x - 3)(x + 3)}{x - 3}) algebraically, identify domain restrictions, and understand its graphical implications. Precise steps, domain warning, and key learning points for algebra students."]