By trial or synthetic division, find \( x = 1 \) is a root. Factorize:

["SEO-Optimized Article: Using Synthetic Division to Confirm ( x = 1 ) is a Root and Factor Polynomials", "---", "# Using Synthetic Division to Confirm ( x = 1 ) as a Root and Factorize Polynomials", "When studying polynomials, one of the key skills is identifying whether a given value—like ( x = 1 )—is a root, which means the polynomial equals zero when substituted. Synthetic division offers a fast, efficient way to test for roots and simultaneously factor polynomials. In this article, we’ll walk through how to use synthetic division to check if ( x = 1 ) is a root, and how to properly factor the polynomial afterward.", "---", "## Why Confirming a Root Matters", "A root of a polynomial ( P(x) ) is a value ( r ) such that ( P(r) = 0 ). Confirming ( x = 1 ) as a root unlocks powerful factorization tools—specifically, the Factor Theorem, which states that if ( x = r ) is a root, then ( (x - r) ) is a factor of ( P(x) ). This is essential for polynomial simplification and solving equations.", "---", "## Step-by-Step: Applying Synthetic Division to Test ( x = 1 )", "### Step 1: Set Up the Polynomial", "Assume we are working with the polynomial:\n[\nP(x) = x^3 - 6x^2 + 11x - 6\n]\n(You can substitute any polynomial—adjust the coefficients as needed.)", "### Step 2: Perform Synthetic Division with ( x = 1 )", "Write the coefficients:\n[\n\ ext{Coefficients: } \underline{1} \quad -6 \quad 11 \quad -6\n]\nUse ( x = 1 ) as the divider:", "<br/>\n1 | 1 -6 11 -6<br/>\n | 1 -5 6</p>\n<hr/>\n<pre><code> 1 -5 6 0\n</code></pre>\n<p>", "- Bring down the 1\n- Multiply: ( 1 \ imes 1 = 1 ), write under (-6), add: (-6 + 1 = -5)\n- Multiply: ( -5 \ imes 1 = -5 ), write under (11), add: (11 - 5 = 6)\n- Multiply: (6 \ imes 1 = 6 ), write under (-6), add: (-6 + 6 = 0)", "Result: The final row gives coefficients: ( 1\quad -5\quad 6\quad 0 )", "### Step 3: Interpret the Output", "Since the remainder is 0, ( x = 1 ) is a root of ( P(x) ). The numbers in the last row represent the coefficients of a quotient polynomial:\n[\nP(x) = (x - 1)(x^2 - 5x + 6)\n]", "The zero remainder confirms the root; the quotient ( x^2 - 5x + 6 ) is the reduced polynomial.", "---", "## Step 4: Factor the Quotient Polynomial", "Now factor ( x^2 - 5x + 6 ) to fully factor the original polynomial:\n[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "Thus, the complete factorization is:\n[\nP(x) = (x - 1)(x - 2)(x - 3)\n]", "---", "## Key Takeaways", "- Use synthetic division to efficiently test for roots and reduce polynomial degree.\n- A zero remainder confirms ( x = r ) is a root.\n- The quotient polynomial provides a direct route to full factorization.\n- This method is faster and less error-prone than polynomial long division for many problems.", "---", "## Why This Matters for Students and Professionals", "Whether tackling algebra, calculus, or real-world applications involving polynomial modeling, confirming roots via synthetic division helps streamline problem-solving. Accurate factorization unlocks solutions to equations, simplifies expressions, and supports graph analysis.", "---", "## Conclusion", "Testing ( x = 1 ) as a root using synthetic division is a fundamental skill in algebra. By confirming ( x = 1 ) as a root of ( x^3 - 6x^2 + 11x - 6 ), we deduce ( (x - 1) ) is a factor and proceed to fully factor the polynomial using quotient division. Mastering this technique enhances your ability to analyze, solve, and simplify polynomials efficiently.", "---", "Keywords:\nsynthetic division, find roots, factor a polynomial, test if x=1 is a root, factor theorem, polynomial division, algebra tutorial, factorize cubic polynomial, polynomial root search, remainder theorem, root confirmation, polynomial factorization, linear factor, quadratic factor", "Meta Description:\nLearn how to use synthetic division to confirm ( x = 1 ) as a root and factor polynomials. Step-by-step guide with example, ideal for algebra students and educators. Master this essential technique today.", "Related Articles:\n- Polynomial Long Division Explained\n- How to Use the Factor Theorem\n- Solving Cubic Equations Step by Step\n- Synthetic Division vs Polynomial Long Division", "---", "Use synthetic division wisely—your polynomial journeys get simpler!"]