Factor the numerator: \( rac{3x(x + 2)}{3x} \).

["Title: Simplifying Rational Expressions: Factoring the Numerator in ( \frac{3x(x + 2)}{3x} )", "When working with rational expressions like ( \frac{3x(x + 2)}{3x} ), one of the first and most important steps is factor the numerator. Proper factoring allows for simplification and better understanding of the expression’s behavior, especially when reducing common factors in the numerator and denominator.", "---", "### Step 1: Understand the Expression", "The given expression is:", "[\n\frac{3x(x + 2)}{3x}\n]", "Here, both the numerator and the denominator are polynomials. To simplify, we look for common factors in both parts.", "---", "### Step 2: Factor the Numerator", "The numerator is:", "[\n3x(x + 2)\n]", "This is already partially factored, but factoring reveals the core components that can cancel with the denominator:", "- The constant ( 3x ) is factored out as a single factor.\n- The expression inside the parentheses, ( x + 2 ), is linear and irreducible over real numbers.", "So, fully factored, the numerator becomes:", "[\n3x(x + 2)\n]", "This factoring is essential because it exposes what’s shared with the denominator ( 3x ).", "---", "### Step 3: Cancel Common Factors", "With numerator and denominator factored:", "[\n\frac{3x(x + 2)}{3x}\n]", "We see that ( 3x ) appears in both numerator and denominator. As long as ( 3x <br/>\neq 0 ) (i.e., ( x <br/>\neq 0 )), we can safely cancel:", "[\n\frac{\cancel{3x}(x + 2)}{\cancel{3x}} = x + 2\n]", "Thus, the simplified expression is:", "[\nx + 2, \quad x <br/>\ne 0\n]", "---", "### Why Factoring the Numerator Matters", "- Simplification: Factoring reveals common factors, enabling clear reduction of the rational expression.\n- Domain Awareness: Recognizing terms like ( 3x ) in the original expression warns us that ( x = 0 ) is excluded from the domain since division by zero is undefined.\n- Understanding Function Behavior: Seeing the simplified form ( x + 2 ) makes the overall behavior of the original rational expression more transparent.", "---", "### Conclusion", "Factoring the numerator in ( \frac{3x(x + 2)}{3x} ) is a vital first step toward simplification. By identifying and canceling the shared factor ( 3x ), we transform the expression into a simpler, equivalent form: ( x + 2 ), with the important restriction ( x <br/>\ne 0 ).", "Mastering this technique strengthens algebraic fluency and is key to solving more complex rational expressions efficiently.", "---", "Keywords: factor the numerator, simplify rational expression, simplify ( \frac{3x(x+2)}{3x} ), cancel common factors, algebraic simplification", "Meta Description: Learn how to factor ( 3x(x + 2) ) and simplify ( \frac{3x(x + 2)}{3x} ) step by step. Discover key rules for rational expressions and domain considerations."]