To find the roots, solve \( x^3 - 6x^2 + 11x - 6 = 0 \).

["# How to Find the Roots of the Equation: ( x^3 - 6x^2 + 11x - 6 = 0 )", "Solving cubic equations is a fundamental skill in algebra, and finding the roots of ( x^3 - 6x^2 + 11x - 6 = 0 ) is a perfect example to explore the process systematically. Whether you're a student tackling quadratic and cubic equations or a self-learner eager to master polynomial solving, this guide will walk you through solving the cubic equation step-by-step.", "## Understanding the Equation", "The given equation is a cubic polynomial:", "[\nx^3 - 6x^2 + 11x - 6 = 0\n]", "By the fundamental theorem of algebra, every cubic polynomial has exactly three roots (real or complex), counting multiplicity. For this equation, we seek real roots that satisfy the equality.", "## Step 1: Rational Root Theorem", "One effective method to find roots of polynomials with integer coefficients is the Rational Root Theorem. This theorem states that any rational solution ( x = \frac{p}{q} ) must have ( p ) dividing the constant term and ( q ) dividing the leading coefficient.", "Here:\n- Constant term = ( -6 ) → Possible ( p ): ( \pm1, \pm2, \pm3, \pm6 )\n- Leading coefficient = 1 → Possible ( q ): ( \pm1 )", "So, possible rational roots are:\n[\n\pm1,\ \pm2,\ \pm3,\ \pm6\n]", "## Step 2: Test Possible Roots", "We substitute these values into the polynomial to test which ones satisfy the equation ( x^3 - 6x^2 + 11x - 6 = 0 ).", "- ( x = 1 ):\n ( 1^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 )\n ✅ ( x = 1 ) is a root!", "Since ( x = 1 ) is a root, ( (x - 1) ) is a factor of the cubic polynomial.", "## Step 3: Polynomial Division (Synthetic or Long Division)", "Now divide ( x^3 - 6x^2 + 11x - 6 ) by ( x - 1 ) to reduce it to a quadratic.", "Using synthetic division:", "[\n\begin{array}{r|rrrr}\n1 & 1 & -6 & 11 & -6 \\n & & 1 & -5 & 6 \\n\hline\n & 1 & -5 & 6 & 0 \\n\end{array}\n]", "The quotient is:\n[\nx^2 - 5x + 6\n]", "So,\n[\nx^3 - 6x^2 + 11x - 6 = (x - 1)(x^2 - 5x + 6)\n]", "## Step 4: Solve the Quadratic Factor", "Now solve ( x^2 - 5x + 6 = 0 ). Factor the quadratic:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "Setting each factor to zero:\n[\nx = 2 \quad \ ext{and} \quad x = 3\n]", "## Step 5: Complete the Roots", "Combining all factors:", "[\nx^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)\n]", "So, the solutions are:\n[\nx = 1,\quad x = 2,\quad x = 3\n]", "All roots are real and rational.", "## Conclusion", "Solving ( x^3 - 6x^2 + 11x - 6 = 0 ) reveals that the roots are simply ( 1, 2, ) and ( 3 ). This example demonstrates a powerful algebraic method using the Rational Root Theorem, polynomial factorization, and quadratic solving—all essential tools when tackling cubic equations. With these techniques, you can confidently find all roots of cubic polynomials.", "---", "### Frequently Asked Questions (FAQs)", "Q: Can a cubic equation have complex roots?\nA: Yes, but in this case, all roots are real and rational. Complex roots appear in conjugate pairs when coefficients are real but are not present here.", "Q: Why does the Rational Root Theorem help?\nA: It limits potential rational roots by restricting candidates based on coefficients, making root testing efficient.", "Q: What if the cubic has no rational roots?\nA: Then methods likeCardano’s formula or numerical approximation (e.g., Newton-Raphson) may be needed.", "---", "Keywords: solve ( x^3 - 6x^2 + 11x - 6 = 0 ), cubic equation roots, polynomial factorization, rational root theorem, real roots cubic, algebraic solution, quadratic equation roots, synthetic division.", "---", "Meta Description:\nLearn how to find the roots of ( x^3 - 6x^2 + 11x - 6 = 0 ) step-by-step using factorization and the Rational Root Theorem. Get the exact solution ( x = 1, 2, 3 ) and understand the method."]