Thus, $u^2 - 5u + 4$ factors as $(u - 4)(u - 1)$.

["How to Factor the Quadratic $u^2 - 5u + 4$: A Step-by-Step Guide (and Why It Factors as $(u - 4)(u - 1)$)", "When learning algebra, one of the most fundamental skills is factoring quadratic expressions. Understanding how to factor $u^2 - 5u + 4$ not only builds confidence in algebra but also forms the foundation for solving equations, simplifying expressions, and analyzing polynomial functions. In this article, we’ll explore why $u^2 - 5u + 4$ factors as $(u - 4)(u - 1)$, walk through the step-by-step process, and explain the significance of this elegant factorization.", "### The Expression: $u^2 - 5u + 4$", "At first glance, the quadratic $u^2 - 5u + 4$ appears simple, but factoring it requires a clear understanding of the method—typically using the AC method or trial-and-error with root analysis. This expression is a standard trinomial where the coefficient of $u^2$ is 1, making factoring straightforward compared to more complex forms.", "### Step-by-Step Factoring Process", "1. Identify coefficients: The quadratic is of the form $u^2 + bu + c$, where:\n - $b = -5$ (sum of roots)\n - $c = 4$ (product of roots)", "2. Find two numbers that multiply to $c = 4$ and add to $b = -5$:\n We need integer pairs whose product is 4 and whose sum is -5.\n Possible factor pairs of 4:\n - $1$ and $4$ → sum = 5\n - $-1$ and $-4$ → sum = $-5$ ✅ \nSo, $-1$ and $-4$ satisfy both conditions.", "3. Rewrite the middle term using these numbers:\n Break $-5u$ into $-1u - 4u$:\n $$\n u^2 - 1u - 4u + 4\n $$", "4. Factor by grouping:\n Group terms to factor out common factors:\n $$\n (u^2 - 1u) + (-4u + 4) = u(u - 1) - 4(u - 1)\n $$", "5. Factor out the common binomial $(u - 1)$:\n $$\n (u - 1)(u - 4)\n $$", "However, since multiplication is commutative, we can write the factors in any order:\n$$\n(u - 4)(u - 1)\n$$", "### Why This Factoring Works: The Meaning Behind $(u - 4)(u - 1)$", "The factorization $(u - 4)(u - 1)$ reveals the roots of the quadratic:\n- $u - 4 = 0 \Rightarrow u = 4$\n- $u - 1 = 0 \Rightarrow u = 1$", "These values make the original expression equal to zero, confirming the correctness of the factorization. Importantly, switching the order $(u - 1)(u - 4)$ does not change the product due to commutativity, preserving the mathematical accuracy.", "### Significance and Applications", "Understanding that $u^2 - 5u + 4 = (u - 4)(u - 1)$ is crucial for:\n- Solving equations: Setting $(u - 4)(u - 1) = 0$ gives precise solutions.\n- Simplifying rational expressions: Factoring enables canceling common terms.\n- Graphing parabolas: The roots $(4, 0)$ and $(1, 0)$ indicate where the parabola $y = u^2 - 5u + 4$ intersects the $u$-axis.\n- Algebraic manipulation: Factoring supports polynomial division, expansion, and advanced topics like derivatives and integrals.", "### Final Thoughts", "Mastering how to factor $u^2 - 5u + 4$ into $(u - 4)(u - 1)$ is a key milestone in algebra. It demonstrates the power of logical decomposition, the importance of identifying root relationships, and the elegance of polynomial structure. Whether for academic success or real-world problem solving, this fact remains a vital building block—showing how breaking complex expressions into simpler components leads to deeper understanding and greater proficiency.", "---", "Keywords: factor $u^2 - 5u + 4$, factor quadratic expression, how to factor $u^2 - 5u + 4$, $(u - 4)(u - 1)$, algebra basics, polynomial factoring, roots of quadratic, factoring quadratics step-by-step."]