We perform synthetic division of \( x^3 - 4x^2 + x + 6 \) by \( x + 1 \):

["Synthetic Division of ( x^3 - 4x^2 + x + 6 ) by ( x + 1 ): A Step-by-Step Guide", "When solving polynomial equations or simplifying rational expressions, synthetic division is a fast and efficient technique, especially when dividing by linear factors. In this article, we’ll explore how to perform synthetic division of the cubic polynomial ( x^3 - 4x^2 + x + 6 ) by the linear divisor ( x + 1 ), step by step. Understanding this process not only helps in polynomial division but also in factoring polynomials and finding roots.", "---", "### What is Synthetic Division?", "Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form ( x - c ). It simplifies the polynomial division procedure, making it quicker and less error-prone compared to long division—particularly useful for cubic and higher-degree polynomials.", "In this guide, we divide:", "[\nP(x) = x^3 - 4x^2 + x + 6 \quad \ ext{by} \quad x + 1\n]", "Note that ( x + 1 = x - (-1) ), so ( c = -1 ).", "---", "### Step 1: Set Up the Coefficients and Value of ( c )", "Write down the coefficients of ( P(x) ) in descending order of powers:", "[\n\begin{array}{cccc}\n\ ext{Coefficients:} & 1 & -4 & 1 & 6 \\n\ ext{Value of } c: & -1 \\n\end{array}\n]", "---", "### Step 2: Apply Synthetic Division Steps", "Step 2.1: Write ( c = -1 ) left of the setup.", "[\n\begin{array}{r|rrrr}\n-1 & 1 & -4 & 1 & 6 \\n & & & & \\n\end{array}\n]", "Step 2.2: Bring down the leading coefficient (1).", "[\n\begin{array}{r|rrrr}\n-1 & 1 & -4 & 1 & 6 \\n & & & & \\n & 1 & & & \\n\end{array}\n]", "Step 2.3: Multiply (-1) by the brought-down value (1), write the result (−1) under the next coefficient (−4).", "[\n-1 \ imes 1 = -1\n]", "[\n\begin{array}{r|rrrr}\n-1 & 1 & -4 & 1 & 6 \\n & & -1 & & \\n & 1 & -1 & & \\n\end{array}\n]", "Step 2.4: Add (-4 + (-1) = -5", "[\n\begin{array}{r|rrrr}\n-1 & 1 & -4 & 1 & 6 \\n & & -1 & & \\n & 1 & -5 & & \\n\end{array}\n]", "Step 2.5: Multiply (-1) by (-5 = 5), write under next coefficient (1).", "[\n-1 \ imes -5 = 5\n]", "[\n\begin{array}{r|rrrr}\n-1 & 1 & -4 & 1 & 6 \\n & & -1 & 5 & \\n & 1 & -5 & 6 & \\n\end{array}\n]", "Step 2.6: Add (1 + 5 = 6", "[\n\begin{array}{r|rrrr}\n-1 & 1 & -4 & 1 & 6 \\n & & -1 & 5 & \\n & 1 & -5 & 6 & 10 \\n\end{array}\n]", "Step 2.7: Multiply (-1 \ imes 6 = -6), write under last coefficient.", "[\n-1 \ imes 6 = -6\n]", "[\n\begin{array}{r|rrrr}\n-1 & 1 & -4 & 1 & 6 \\n & & -1 & 5 & -6 \\n & 1 & -5 & 6 & 10 \\n\end{array}\n]", "Step 2.8: Add (6 + (-6) = 0)", "Since the remainder is 0, ( x + 1 ) is a factor of ( P(x) ).", "---", "### Step 3: Interpret the Result", "The synthetic division process gives:", "- The quotient polynomial is:\n [\n \boxed{x^2 - 5x + 6}\n ]\n (degree one less than original cubic, as expected.)", "- The remainder is:\n [\n \boxed{0}\n ]\n confirming perfect divisibility.", "---", "### Step 4: Write the Final Result", "Since the remainder is zero, we conclude:", "[\nx^3 - 4x^2 + x + 6 = (x + 1)(x^2 - 5x + 6)\n]", "We can further factor the quadratic:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "Thus,", "[\nx^3 - 4x^2 + x + 6 = (x + 1)(x - 2)(x - 3)\n]", "---", "### Why Synthetic Division Matters", "- Speed and Simplification: Quickly divide polynomials without writing full long division.\n- Finding Roots: If synthetic division yields zero remainder, ( x + 1 = 0 \Rightarrow x = -1 ) is a root.\n- Factorization: Helps break down higher-degree polynomials into products of linear and irreducible quadratics.", "---", "### Summary", "Synthetic division of ( x^3 - 4x^2 + x + 6 ) by ( x + 1 ) gave us a quotient of ( x^2 - 5x + 6 ) and remainder 0. This confirms:", "[\nx^3 - 4x^2 + x + 6 = (x + 1)(x^2 - 5x + 6)\n]", "Understanding and practicing synthetic division equips you with a powerful tool for polynomial analysis and equation solving.", "---", "Keywords: synthetic division, polynomial division, ( x^3 - 4x^2 + x + 6 ), ( x + 1 ), factor theorem, polynomial factorization, cubic polynomial, short division method.", "Meta Description: Learn how to perform synthetic division of ( x^3 - 4x^2 + x + 6 ) by ( x + 1 ), step by step, with remainder analysis and factorization results.", "---", "Disclaimer: This guide is intended for educational purposes. Synthetic division applies only to linear divisors of the form ( x - c )."]