A function is defined by \( f(x) = x^3 - 6x^2 + 11x - 6 \). Find the x-values where the function crosses the x-axis.
["Title: Find the x-Intercepts of ( f(x) = x^3 - 6x^2 + 11x - 6 ): A Step-by-Step Guide", "Introduction\nUnderstanding where a function crosses the x-axis is essential in algebra and calculus. These points, known as the x-intercepts or roots, reveal critical information about the behavior of polynomial functions. In this article, we’ll analyze the cubic function ( f(x) = x^3 - 6x^2 + 11x - 6 ) and pinpoint the exact x-values where the graph intersects the x-axis.", "---", "### What Does It Mean for a Function to Cross the X-Axis?\nA function ( f(x) ) crosses the x-axis where ( f(x) = 0 ). The solutions to the equation\n[ x^3 - 6x^2 + 11x - 6 = 0 ]\nare the roots of the polynomial, which correspond to the x-values of intersection points.", "---", "### Step 1: Identify Possible Rational Roots\nSince this is a cubic polynomial with integer coefficients, we apply the Rational Root Theorem, which states that any rational root ( \frac{p}{q} ) must have ( p ) as a factor of the constant term (-6) and ( q ) as a factor of the leading coefficient (1).", "Factors of -6: ( \pm1, \pm2, \pm3, \pm6 )\nAmong these, ( q = 1 ), so possible rational roots are:\n[ \pm1, \pm2, \pm3, \pm6 ]", "We test these values by substituting into ( f(x) ):", "- ( f(1) = 1^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 ) ✅\nSo, ( x = 1 ) is a root.", "---", "### Step 2: Use Polynomial Division or Factorization\nSince ( x = 1 ) is a root, ( (x - 1) ) is a factor. We perform polynomial division or synthetic division to factor out ( (x - 1) ) from ( f(x) ).", "Using synthetic division:", "1 | 1 -6 11 -6 \n | 1 -5 6 \n ─────────────────────── \n 1 -5 6 0", "The quotient polynomial is ( x^2 - 5x + 6 ). Thus,\n[ f(x) = (x - 1)(x^2 - 5x + 6) ]", "---", "### Step 3: Factor the Quadratic\nNow factor ( x^2 - 5x + 6 ):\nLooking for two numbers that multiply to ( 6 ) and add to ( -5 ), we find ( -2 ) and ( -3 ).\nSo,\n[ x^2 - 5x + 6 = (x - 2)(x - 3) ]", "Therefore, the full factorization is:\n[ f(x) = (x - 1)(x - 2)(x - 3) ]", "---", "### Step 4: Solve for the Roots\nSet each factor equal to zero:\n[ x - 1 = 0 \Rightarrow x = 1 ]\n[ x - 2 = 0 \Rightarrow x = 2 ]\n[ x - 3 = 0 \Rightarrow x = 3 ]", "---", "### Final Answer: The x-Values Where the Function Crosses the X-Axis\nThe function ( f(x) = x^3 - 6x^2 + 11x - 6 ) crosses the x-axis at:\n[ \boxed{x = 1,\ 2,\ 3} ]", "---", "### Summary\nBy applying the Rational Root Theorem and factoring, we determined the exact x-intercepts of the cubic function. These roots are not only mathematically significant but also useful in graphing, optimization, and real-world modeling. Whether you're solving equations or analyzing curves, understanding how to find x-intercepts is a foundational skill in mathematics.", "---", "Keywords:\nx-intercepts, roots of a function, solve ( f(x) = 0 ), polynomial factorization, rational root theorem, ( f(x) = x^3 - 6x^2 + 11x - 6 ), function analysis, algebraic roots\nMeta Description:\nFind the exact x-values where ( f(x) = x^3 - 6x^2 + 11x - 6 ) crosses the x-axis by factoring and solving using the Rational Root Theorem.\nTags: #Algebra #Polynomials #FunctionGraphing #Roots #MathHelp #STEMEducation"]
