["SEO Article: Solve the Equation $ u^3 - 5u^2 + 6u = 0 $ — Step-by-Step Factoring and Solutions", "Understanding how to factor polynomial equations is essential for solving them efficiently. One commonly encountered problem is the cubic equation:", "$$
\nu^3 - 5u^2 + 6u = 0
\n$$", "In this article, we’ll break down the factoring process, solve for $ u $, and explain why this approach is valuable in algebra and related fields.", "---", "### Why Factor Polynomial Equations?", "Factoring transforms complex polynomial equations into simpler, multiplying terms. This makes it easier to find the roots—the values of $ u $ that satisfy the equation:", "$$
\nu(u^2 - 5u + 6) = 0
\n$$", "From here, the equation becomes clear: either $ u = 0 $ or the quadratic $ u^2 - 5u + 6 = 0 $. Factoring enables us to solve such equations quickly and accurately.", "---", "### Step-by-Step Factoring Process", "Start with the original equation:", "$$
\nu^3 - 5u^2 + 6u = 0
\n$$", "#### Step 1: Factor out the common term $ u $", "$$
\nu(u^2 - 5u + 6) = 0
\n$$", "Now, we analyze the quadratic expression inside the parentheses: $ u^2 - 5u + 6 $.", "#### Step 2: Factor the quadratic trinomial", "Look for two numbers that multiply to $ +6 $ and add to $ -5 $. These numbers are $ -2 $ and $ -3 $, because:", "$$
\n(-2) \ imes (-3) = 6 \quad \ ext{and} \quad (-2) + (-3) = -5
\n$$", "Thus,", "$$
\nu^2 - 5u + 6 = (u - 2)(u - 3)
\n$$", "#### Step 3: Write the fully factored form", "$$
\nu(u - 2)(u - 3) = 0
\n$$", "---", "### Solve for $ u $", "Set each factor equal to zero:", "1. $ u = 0 $
\n2. $ u - 2 = 0 \Rightarrow u = 2 $
\n3. $ u - 3 = 0 \Rightarrow u = 3 $", "---", "### Final Answer", "The solutions to the equation $ u^3 - 5u^2 + 6u = 0 $ are:", "$$
\nu = 0, \quad u = 2, \quad u = 3
\n$$", "---", "### Summary", "Factoring $ u^3 - 5u^2 + 6u $ is straightforward once you recognize the shared factor $ u $ and factor the resulting quadratic:", "$$
\nu^3 - 5u^2 + 6u = u(u - 2)(u - 3)
\n$$", "This factoring reveals the roots clearly and enables quick solution finding—key skills in algebra, calculus, and engineering applications.", "---", "Keywords: factor $ u^3 - 5u^2 + 6u $, solve cubic equation, factor trinomial $ u^2 - 5u + 6 $, polynomial roots, algebraic factoring, step-by-step solution, math tutorial.", "Meta Description: Learn how to factor and solve $ u^3 - 5u^2 + 6u = 0 $ using step-by-step factoring, including finding all roots and verifying solutions mathematically."]
