\(x^3 - 3x^2 + 2x = x((5x - 6) - 3x + 2) = x(2x - 4) = 2x^2 - 4x\).
["# Solving and Factoring the Polynomial Equation: (x^3 - 3x^2 + 2x = x((5x - 6) - 3x + 2) = 2x^2 - 4x)", "Understanding polynomial equations is fundamental in algebra, especially when simplifying complex expressions. One useful example is transforming and factoring the cubic expression:", "[\nx^3 - 3x^2 + 2x = x((5x - 6) - 3x + 2) = 2x^2 - 4x\n]", "In this guide, we’ll walk through the step-by-step simplification, expansion, and factoring of this expression to reveal its underlying structure.", "---", "## Step 1: Simplify the Left-Hand Side Polynomial", "Start with the original cubic polynomial:", "[\nx^3 - 3x^2 + 2x\n]", "Factor out the common term (x):", "[\nx(x^2 - 3x + 2)\n]", "Now factor the quadratic:", "[\nx^2 - 3x + 2 = (x - 1)(x - 2)\n]", "So the fully factored form of the left-hand side is:", "[\nx(x - 1)(x - 2)\n]", "---", "## Step 2: Simplify the Right-Hand Side Expression", "The right-hand side of the original identity is:", "[\nx((5x - 6) - 3x + 2)\n]", "Simplify inside the parentheses:", "[\n(5x - 6 - 3x + 2) = (2x - 4)\n]", "Multiply by (x):", "[\nx(2x - 4) = 2x^2 - 4x\n]", "---", "## Step 3: Equating Both Sides", "We now have:", "[\nx^3 - 3x^2 + 2x = 2x^2 - 4x\n]", "Bring all terms to one side:", "[\nx^3 - 3x^2 + 2x - (2x^2 - 4x) = 0\n]", "Simplify:", "[\nx^3 - 3x^2 + 2x - 2x^2 + 4x = x^3 - 5x^2 + 6x = 0\n]", "Factor out (x):", "[\nx(x^2 - 5x + 6) = 0\n]", "Factor the quadratic:", "[\nx(x - 2)(x - 3) = 0\n]", "This confirms the solutions: (x = 0), (x = 2), and (x = 3), matching the roots (x = 0), (x = 1), and (x = 2) found from the factored form.", "---", "## Step 4: Understanding the Factored Form", "From earlier, we learned:", "[\nx^3 - 3x^2 + 2x = x(x - 1)(x - 2) \quad \ ext{and} \quad 2x^2 - 4x = 2x(x - 2)\n]", "Although the expressions look different in expanded form, factoring reveals their equivalence in different representations. The key insight is recognizing how factoring simplifies solving and analyzing polynomial behavior.", "---", "## Why Factoring Matters", "Factoring transforms complex expressions into products of simpler terms, making it easier to:", "- Solve equations by setting each factor to zero\n- Graph polynomial behavior through its roots\n- Simplify rational expressions and partial fractions", "In this example, matching both original sides using factoring verifies the algebraic equivalence:", "[\nx(x - 1)(x - 2) = 2x(x - 2)\n]", "Dividing both sides by (x(x - 2)) (where (x <br/>\neq 0, 2)) gives:", "[\nx - 1 = 2 \quad \Rightarrow \quad x = 3\n]", "But note that (x = 0) and (x = 2) are common roots, showing shared solutions.", "---", "## Conclusion", "Factoring and equivalence checking simplify and validate polynomial expressions. Through step-by-step simplification from (x^3 - 3x^2 + 2x) to (2x^2 - 4x), we uncovered deeper structure, factored forms, and solution insights.", "Mastering these skills empowers deeper algebraic fluency and effective problem-solving in higher mathematics.", "---", "## SEO Keywords:\nx³ – 3x² + 2x, factor polynomial, solve cubic equation, simplify algebra, rational equation, polynomial identity, factoring techniques, algebraic simplification, roots of polynomials, factoring expressions, polynomial factorization, algebraic equations.", "---", "### Practical Tip\nAlways verify equivalency by substituting roots back into the original and simplified expressions to confirm correctness — especially after expanding or factoring complex polynomials.", "---", "\nExplore more algebra tutorials and polynomial simplification guides to strengthen your math foundation!"]
