1. **Problem Statement:** Perform synthetic division of a polynomial by a linear divisor of the form $x - c$. 2. **Formula and Rules:** Synthetic division is a shortcut method for dividing a polynomial $P(x)$ by $x - c$. The key steps involve using the root $c$ and the coefficients of $P(x)$. 3. **Step-by-step process:** - Write down the coefficients of the polynomial. - Bring down the leading coefficient. - Multiply it by $c$ and write the result under the next coefficient. - Add the column and repeat until all coefficients are processed. 4. **Example:** Divide $2x^3 - 6x^2 + 2x - 1$ by $x - 3$. - Coefficients: 2, -6, 2, -1 - Root $c = 3$ 5. **Perform synthetic division:** \begin{align*} &\text{Bring down } 2 \ &2 \times 3 = 6 \quad \Rightarrow -6 + 6 = 0 \ &0 \times 3 = 0 \quad \Rightarrow 2 + 0 = 2 \ &2 \times 3 = 6 \quad \Rightarrow -1 + 6 = 5 \end{align*} 6. **Result:** The quotient polynomial is $2x^2 + 0x + 2 = 2x^2 + 2$ and the remainder is $5$. 7. **Final answer:** $$\frac{2x^3 - 6x^2 + 2x - 1}{x - 3} = 2x^2 + 2 + \frac{5}{x - 3}$$