rac{x^3 - 3x}{x^2 + 1} = x \Rightarrow x^3 - 3x = x(x^2 + 1) = x^3 + x \Rightarrow -4x = 0 \Rightarrow x = 0

["# Solving the Equation: ( \frac{x^3 - 3x}{x^2 + 1} = x ) Step by Step", "Solving algebraic equations step-by-step is essential for building strong mathematical comprehension, especially when dealing with rational expressions. In this article, we will carefully analyze and solve the equation:", "[\n\frac{x^3 - 3x}{x^2 + 1} = x\n]", "Understanding how to manipulate and simplify such equations helps clarify underlying algebraic principles and avoids common pitfalls.", "## Step 1: Eliminate the Denominator", "To simplify the rational expression, multiply both sides of the equation by ( x^2 + 1 ), since ( x^2 + 1 <br/>\neq 0 ) for all real numbers ( x ). This step removes the denominator:", "[\nx^3 - 3x = x(x^2 + 1)\n]", "This transformation is valid as ( x^2 + 1 > 0 ) always, ensuring no division by zero.", "## Step 2: Expand the Right-Hand Side", "Now expand the right-hand side of the equation:", "[\nx(x^2 + 1) = x^3 + x\n]", "Substitute back into the equation:", "[\nx^3 - 3x = x^3 + x\n]", "## Step 3: Simplify Both Sides", "Subtract ( x^3 ) from both sides:", "[\n-3x = x\n]", "Next, subtract ( x ) from both sides:", "[\n-3x - x = 0 \quad \Rightarrow \quad -4x = 0\n]", "## Step 4: Solve for ( x )", "Divide both sides by (-4):", "[\nx = 0\n]", "## Verifying the Solution", "Since ( x = 0 ), substitute back into the original equation:", "[\n\frac{0^3 - 3(0)}{0^2 + 1} = \frac{0}{1} = 0 \quad \ ext{and} \quad x = 0\n]", "The equation holds true: ( 0 = 0 ), confirming the solution.", "## Why This Approach Works", "By eliminating the denominator first and then simplifying algebraically, we avoid complicated expressions and isolate ( x ) cleanly. Each step preserves the equation’s equality, allowing clear deduction without introducing extraneous solutions.", "---", "### Conclusion", "Solving ( \frac{x^3 - 3x}{x^2 + 1} = x ) leads directly to ( x = 0 ) after logical simplification. This process demonstrates careful algebraic manipulation, valid domain considerations, and verification of solutions—essential skills for mastering rational equations.", "Search terms: solve rational equation step by step, simplify rational expression, how to solve x³ - 3x = x(x² + 1), solve x³ - 3x = x(x² + 1)"]