Question: Factor the expression $12x^3 - 18x^2 + 6x$ completely.
["Factor the Expression $12x^3 - 18x^2 + 6x$ Completely: A Step-by-Step Guide", "Understanding how to factor polynomial expressions is essential for mastering algebra. One of the most frequently encountered expressions in foundational math is:\n$$12x^3 - 18x^2 + 6x$$\nWhether you're a student preparing for exams or a teacher seeking clear examples, learning to factor this expression completely helps build critical problem-solving skills. In this article, we’ll walk through the complete factoring process and explain each step clearly.", "---", "### Step 1: Identify the Greatest Common Factor (GCF)", "The first step in factoring any polynomial is identifying the Greatest Common Factor (GCF) of all the terms.", "The given expression is:\n$$\n12x^3 - 18x^2 + 6x\n$$", "Let’s factor each coefficient and variable term:\n- Coefficients: GCF of 12, 18, and 6 is 6\n- Variables: The smallest power of $x$ present in all terms is $x^1$", "So, the GCF is:\n$$\n6x\n$$", "Now factor out $6x$ from every term:\n$$\n12x^3 = 6x \cdot 2x^2\n$$\n$$\n-18x^2 = 6x \cdot (-3x)\n$$\n$$\n6x = 6x \cdot 1\n$$", "Rewriting the expression:\n$$\n12x^3 - 18x^2 + 6x = 6x(2x^2 - 3x + 1)\n$$", "---", "### Step 2: Factor the Quadratic Expression $2x^2 - 3x + 1$", "We now focus on factoring the quadratic inside the parentheses:\n$$\n2x^2 - 3x + 1\n$$", "We use the AC method: Multiply the leading coefficient and constant term:\n$$\nAC = 2 \ imes 1 = 2\n$$", "Look for two numbers that multiply to $AC = 2$ and add up to the middle coefficient $-3$.", "The numbers $-2$ and $-1$ satisfy:\n$$\n(-2) \ imes (-1) = 2 \quad \ ext{and} \quad -2 + (-1) = -3\n$$", "Now split the middle term using these numbers:\n$$\n2x^2 - 2x - x + 1\n$$", "Group terms:\n$$\n(2x^2 - 2x) + (-x + 1)\n$$", "Factor out the GCF from each group:\n$$\n2x(x - 1) -1(x - 1)\n$$", "Now factor out the common binomial factor $(x - 1)$:\n$$\n(2x - 1)(x - 1)\n$$", "---", "### Step 3: Write the Complete Factored Form", "Substituting this back into the earlier result:\n$$\n12x^3 - 18x^2 + 6x = 6x(2x - 1)(x - 1)\n$$", "This is the fully factored form.", "---", "### Why Factor This Expression?", "Factoring expressions like $12x^3 - 18x^2 + 6x$ improves algebraic fluency and is crucial for solving equations, simplifying rational expressions, and analyzing polynomial behavior. The expression reduces neatly due to a clear GCF and a well-structured trinomial.", "---", "### Final Answer:\n$$\n\boxed{12x^3 - 18x^2 + 6x = 6x(2x - 1)(x - 1)}\n$$", "---", "Keywords for SEO:\n- Factor $12x^3 - 18x^2 + 6x$\n- Polynomial factoring\n- Algebraic factoring guide\n- Complete factoring steps\n- Common factor and trinomial factoring", "Mastering these steps empowers you to tackle similar expressions with confidence—starting with factoring out the GCF, identifying special patterns, and fully simplifying to lowest terms."]
