We solve \( p(u) = u^3 - 3u^2 + 2u = 0 \).

["# How to Solve the Cubic Equation ( p(u) = u^3 - 3u^2 + 2u = 0 )", "Solving polynomial equations is a fundamental skill in algebra and plays a crucial role in many areas of mathematics, engineering, and applied sciences. One commonly encountered problem is finding the roots of the cubic equation:", "[\np(u) = u^3 - 3u^2 + 2u = 0\n]", "If you're asking, “How do we solve ( p(u) = 0 )?”, this article provides a clear, step-by-step guide to solving this cubic equation exactly, understanding the logic behind each step, and applying the solution in real-world contexts.", "---", "## Step 1: Factor Out the Common Term", "The first and most straightforward method is factoring. Observe that ( u ) is common in every term:", "[\nu^3 - 3u^2 + 2u = u(u^2 - 3u + 2)\n]", "Now set the factored expression equal to zero:", "[\nu(u^2 - 3u + 2) = 0\n]", "This product equals zero only when at least one factor is zero. So, we split the equation into two parts:", "[\nu = 0 \quad \ ext{or} \quad u^2 - 3u + 2 = 0\n]", "---", "## Step 2: Solve the Quadratic Factor", "Now focus on solving the quadratic equation:", "[\nu^2 - 3u + 2 = 0\n]", "This quadratic can be factored directly:", "[\nu^2 - 3u + 2 = (u - 1)(u - 2)\n]", "So the full factorization of ( p(u) ) is:", "[\np(u) = u(u - 1)(u - 2)\n]", "Setting this equal to zero gives:", "[\nu = 0, \quad u = 1, \quad u = 2\n]", "---", "## Step 3: State the Final Solution", "The solutions to ( p(u) = u^3 - 3u^2 + 2u = 0 ) are:", "[\n\boxed{u = 0,\quad u = 1,\quad u = 2}\n]", "These three values are the roots of the equation — the points where the cubic polynomial intersects the ( u )-axis.", "---", "## Why This Matters: Real-World Applications", "Understanding how to solve equations like ( u^3 - 3u^2 + 2u = 0 ) is essential in various fields, including:", "- Physics: Modeling motion, oscillations, or equilibrium points where systems balance (roots represent stable or unstable points).\n- Engineering: Designing circuits, control systems, or structural balances.\n- Economics: Analyzing equilibrium points in supply-demand models or cost functions.", "---", "## Summary: Key Methods to Solve ( p(u) = 0 )", "| Method | Description | Result |\n|------------------------|-----------------------------------------------------|-----------------------------|\n| Factoring | Extract common term, factor completely | Direct roots found easily |\n| Quadratic Formula | Apply ( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) | Alternative when factoring fails |\n| Graphical Analysis | Plot function to visually identify roots | Useful for deeper insight |", "In this case, factoring was efficient due to the simple structure of ( p(u) ). But for more complex cubics without obvious factors, combining factoring with the quadratic formula is essential.", "---", "## Want to Practice? Try This:", "Solve ( v(v^2 - 5v + 6) = 0 ). What are all possible values of ( v )?", "Hint: Factor the quadratic, then apply the zero product property.", "---", "Conclusion: Solving ( u^3 - 3u^2 + 2u = 0 ) showcases essential algebraic techniques — factoring, the zero product property, and quadratic analysis. Mastering these tools unlocks deeper mathematical understanding and practical problem-solving skills.", "---", "Keywords: Solve ( u^3 - 3u^2 + 2u = 0 ), cubic equation roots, factoring polynomial, zero product property, algebra solution method, real roots of cubic, cubic equation guide", "Meta Description: Learn how to solve ( u^3 - 3u^2 + 2u = 0 ) step-by-step using factoring and root-finding techniques. Key methods explained with working solutions."]