To solve the inequality \( x^3 - 4x^2 + x + 6 < 0 \), we first find the roots of the cubic polynomial \( x^3 - 4x^2 + x + 6 = 0 \).

["# How to Solve the Inequality ( x^3 - 4x^2 + x + 6 < 0 ): Finding Roots of the Cubic Polynomial", "Solving a cubic inequality like ( x^3 - 4x^2 + x + 6 < 0 ) can feel daunting at first, but understanding the core step—finding the roots of the corresponding cubic equation—makes the problem manageable. In this article, we walk through the full process using key algebraic techniques, including factoring, synthetic division, and sign analysis.", "---", "## Step 1: Solve the Corresponding Cubic Equation", "To solve ( x^3 - 4x^2 + x + 6 < 0 ), we begin by finding the roots of the equation:", "[\nx^3 - 4x^2 + x + 6 = 0\n]", "This cubic polynomial may not factor obviously at first glance, so we search for rational roots using the Rational Root Theorem. Possible rational roots are the divisors of the constant term (6) divided by the leading coefficient (1), so possible candidates are:", "[\n\pm1, \pm2, \pm3, \pm6\n]", "Let’s test these values in the polynomial:", "- ( x = 1 ):\n ( 1^3 - 4(1)^2 + 1 + 6 = 1 - 4 + 1 + 6 = 4 <br/>\ne 0 )\n- ( x = -1 ):\n ( (-1)^3 - 4(-1)^2 + (-1) + 6 = -1 - 4 - 1 + 6 = 0 ) ✅", "So, ( x = -1 ) is a root. This means ( (x + 1) ) is a factor.", "---", "## Step 2: Factor the Cubic Polynomial", "Now that ( x = -1 ) is a root, we perform polynomial division or synthetic division to factor out ( (x + 1) ) from ( x^3 - 4x^2 + x + 6 ).", "Using synthetic division:", "-1 | 1 -4 1 6\n | -1 5 -6\n ----------------\n 1 -5 6 0", "The quotient is ( x^2 - 5x + 6 ), so:", "[\nx^3 - 4x^2 + x + 6 = (x + 1)(x^2 - 5x + 6)\n]", "Next, factor the quadratic:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "Thus, the full factorization is:", "[\nx^3 - 4x^2 + x + 6 = (x + 1)(x - 2)(x - 3)\n]", "---", "## Step 3: Determine the Roots", "Set each factor equal to zero:", "[\nx + 1 = 0 \Rightarrow x = -1\n]\n[\nx - 2 = 0 \Rightarrow x = 2\n]\n[\nx - 3 = 0 \Rightarrow x = 3\n]", "So the roots are ( x = -1,\ 2,\ 3 ). These values divide the real number line into four intervals:", "[\n(-\infty, -1),\ (-1, 2),\ (2, 3),\ (3, \infty)\n]", "---", "## Step 4: Analyze the Sign of the Polynomial on Each Interval", "The sign of ( x^3 - 4x^2 + x + 6 ) changes across these intervals at each root, but not necessarily at the roots themselves (since it’s a strict inequality ( < 0 )). To determine where the expression is negative, we test a point in each interval.", "- Interval ( (-\infty, -1) ): Choose ( x = -2 )\n ( (-2 + 1)(-2 - 2)(-2 - 3) = (-1)(-4)(-5) = -20 < 0 ) ✅", "- Interval ( (-1, 2) ): Choose ( x = 0 )\n ( (0 + 1)(0 - 2)(0 - 3) = (1)(-2)(-3) = 6 > 0 ) ❌", "- Interval ( (2, 3) ): Choose ( x = 2.5 )\n ( (3.5)(0.5)(-0.5) = 3.5 \cdot 0.5 \cdot (-0.5) = -0.875 < 0 ) ✅", "- Interval ( (3, \infty) ): Choose ( x = 4 )\n ( (5)(2)(1) = 10 > 0 ) ❌", "The inequality is satisfied where the expression is negative:", "[\nx \in (-\infty, -1) \cup (2, 3)\n]", "---", "## Why This Works", "By finding the exact roots and analyzing sign changes across interval boundaries, we efficiently solve cubic inequalities without graphing. This method leverages algebraic factoring, rational root testing, and strategic testing—key SEO keywords like “how to solve cubic inequalities factoring” and “find roots of cubic polynomial” align naturally with user search intent.", "---", "## Final Answer", "[\n\boxed{x^3 - 4x^2 + x + 6 < 0 \quad \ ext{is satisfied for} \quad x \in (-\infty, -1) \cup (2, 3)}\n]", "Understanding the root-finding process not only solves the inequality but strengthens foundational algebra skills critical for advanced math and competition preparation. Use these techniques to tackle similar challenges efficiently and accurately!", "---", "SEO Keywords: \nSolve (x^3 - 4x^2 + x + 6 < 0), cubic inequality solving, find roots of cubic polynomial, rational root theorem, sign analysis inequality, polynomial factoring, algebra 3 textbook solutions", "Meta Description:\nLearn how to solve (x^3 - 4x^2 + x + 6 < 0) by finding its real roots through factoring, testing intervals, and sign analysis—perfect for students mastering cubic inequalities."]