Divide $ R(x) $ by $ x - 1 $ using synthetic division:

["# Divide $ R(x) $ by $ x - 1 $ Using Synthetic Division: A Step-by-Step Guide", "Understanding polynomial division is fundamental in algebra, especially when evaluating polynomials, finding roots, and simplifying expressions. One of the most efficient methods for dividing a polynomial $ R(x) $ by a linear divisor $ x - c $ is synthetic division. In this article, we’ll explore how to divide $ R(x) $ by $ x - 1 $ using synthetic division, why it works, and how to interpret the result.", "---", "## What is Synthetic Division?", "Synthetic division is a shorthand algorithm used to divide a polynomial $ R(x) $ by a linear divisor of the form $ x - c $. Unlike traditional polynomial long division, synthetic division is faster and simpler—especially for divisors where $ c $ is a small integer or zero. It relies on polynomial identities and eliminates many unnecessary term-by-term subtractions.", "---", "## When to Use $ x - 1 $ as the Divisor", "When dividing $ R(x) $ by $ x - 1 $, we set $ c = 1 $. This makes synthetic division efficient because each step involves multiplying by 1, so no extra coefficients appear — simplifying the calculation.", "---", "## Step-by-Step Synthetic Division of $ R(x) $ by $ x - 1 $", "Let’s walk through the process with a general polynomial:", "Let\n$$\nR(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\n$$", "We want to compute $ R(x) \div (x - 1) $ using synthetic division with $ c = 1 $.", "### Step 1: Write the coefficients of $ R(x) $", "Arrange the coefficients of $ R(x) $ in descending order:\n$$\n[a_n,\ a_{n-1},\ \ldots,\ a_2,\ a_1,\ a_0]\n$$", "### Step 2: Set up the synthetic division box", "Write $ c = 1 $ on the left:", "1 |\na_n a_{n-1} a_{n-2} ... a_2 a_1 a_0", "### Step 3: Bring down the leading coefficient", "Bring down $ a_n $ directly below the line:", "1 |\na_n a_{n-1} a_{n-2} ... a_2 a_1 a_0\n |\n └──────────────────────────────────────────", "### Step 4: Multiply and add repeatedly", "For each coefficient to the right:", "- Multiply the value just below the line by $ c = 1 $\n- Write the result under the next coefficient\n- Add to get the new value below the line", "[\n\begin{array}{r|rrrrrr}\n1 & a_n & a_{n-1} & a_{n-2} & a_{n-3} & a_{n-4} & a_1 & a_0 \\n & & a_n & a_n + a_{n-1} & a_n + (a_{n-1} + a_{n-2}) & \cdots \ \hline\n & a_n & a_n + a_{n-1} & a_n + a_{n-1} + a_{n-2} & \cdots & a_0 + \ ext{current sum} \\n\end{array}\n]", "Eventually, the final number below the line is the remainder of the division.", "---", "## Example: Synthetic Division of $ R(x) = 2x^4 - 3x^3 + 5x - 7 $ by $ x - 1 $", "### Set up:\n$$\nc = 1,\quad R(x) = 2x^4 + 0x^3 + 0x^2 - 3x + 5 - 7\n$$", "Coefficients: $ [2,\ 0,\ 0,\ -3,\ 5,\ -7] $", "Set up:", "1 | 2 0 0 -3 5 -7\n |\n └──────────────────────────────────────────", "Step-by-step:", "1. Bring down $ 2 $ \n2\n", "2. Multiply: $ 1 \ imes 2 = 2 $, write under $ 0 $", "\n2 | 2 0 0 -3 5 -7\n | 2\n └────────────────────────────────────────\n 2\n", "3. Add: $ 0 + 2 = 2 $, write under next \n</code></pre>\n<p>2 | 2 0 0 -3 5 -7<br/>\n | 2<br/>\n └────────────────────────────────────────<br/>\n 2 2<br/>\n<code>", "4. Multiply: $ 1 \ imes 2 = 2 $, add to next $ 0 $", "</code><br/>\n2 | 2 0 0 -3 5 -7<br/>\n | 2 2<br/>\n └────────────────────────────────────────<br/>\n 2 2 2<br/>\n", "5. Multiply: $ 1 \ imes 2 = 2 $, add to $ -3 $ → $ -1 $ \n2 | 2 0 0 -3 5 -7\n | 2 2 2\n └────────────────────────────────────────\n 2 2 2 -1\n", "6. Multiply: $ 1 \ imes -1 = -1 $, add to $ 5 $ → $ 4 $ \n</code></pre>\n<p>2 | 2 0 0 -3 5 -7<br/>\n | 2 2 2 -1<br/>\n └────────────────────────────────────────<br/>\n 2 2 2 -1 4<br/>\n<code>", "7. Multiply: $ 1 \ imes 4 = 4 $, add to $ -7 $ → $ -3 $</code><br/>\n2 | 2 0 0 -3 5 -7<br/>\n | 2 2 2 -1 4<br/>\n └────────────────────────────────────────<br/>\n 2 2 2 -1 4 -3<br/>\n<code>", "### Final synthetic division tableau:", "</code><br/>\n1 | 2 0 0 -3 5 -7<br/>\n | 2 2 2 -1 4 -3<br/>\n └───────────────────────────────────────────<br/>\n 2 2 2 -1 4 -3 r<br/>\n", "### Interpret the result", "- Quotient: Discard the last remainder — the coefficients vary the degree\n The result of $ R(x) \div (x - 1) $ is a polynomial of degree one less than $ R(x) $:\n $$\n Q(x) = 2x^3 + 2x^2 + 2x - 1\n $$", "- Remainder: $ r = -3 $", "So we can write:", "$$\nR(x) = (x - 1)(2x^3 + 2x^2 + 2x - 1) - 3\n$$", "---", "## Why Synthetic Division Works", "Synthetic division relies on the Remainder Theorem, which states that the remainder of dividing $ R(x) $ by $ x - c $ is exactly $ R(c) $. When synthetic division finishes, the final remainder confirms $ R(1) = -3 $.", "This method is especially powerful when factoring polynomials, identifying linear factors, or quickly evaluating polynomials at specific values.", "---", "## Summary", "- Use synthetic division to divide $ R(x) $ by $ x - 1 $ efficiently.\n- Set $ c = 1 $ and follow the step-by-step process.\n- The final remainder is $ R(1) $, and the quotient is a polynomial of degree $ \deg R(x) - 1 $.\n- This method simplifies polynomial division and is essential for root testing and factorization.", "Whether you're solving equations, analyzing graphs, or teaching polynomial division, mastering synthetic division with $ x - 1 $ is a valuable skill in algebra and beyond.", "---", "### Key Takeaways", "- Synthetic division is fast for linear divisors.\n- Use $ c = 1 $ when dividing by $ x - 1 $.\n- The remainder equals $ R(1) $.\n- The quotient skips one degree.\n- Perfect for activating the Remainder Theorem and factor testing.", "---", "Optimize your polynomial calculations with synthetic division — it’s efficient, easy, and widely applicable!"]