1. **Problem:** Use synthetic division to divide a polynomial by a linear binomial. 2. **Step 1: Identify divisor root.** For division by $x - r$, the root is $r$. 3. **Step 2: Write coefficients.** Write down the coefficients of the dividend polynomial in descending order of powers. 4. **Step 3: Setup synthetic division table.** Place root $r$ to the left and draw a horizontal line for the result. 5. **Step 4: Bring down first coefficient.** Bring the first coefficient down as is. 6. **Step 5: Multiply and add repeatedly.** Multiply the root $r$ by the number just written and add to the next coefficient. Repeat this across all coefficients. 7. **Step 6: Interpret result.** The numbers at the bottom (except last) form the coefficients of the quotient polynomial. The last number is the remainder. **Example:** Divide $2x^3 - 3x^2 + 4x - 5$ by $x - 2$. 1. Root is $r=2$. 2. Coefficients: $2, -3, 4, -5$. 3. Setup: | 2 | | | --- | --- | | | 2 | -3 | 4 | -5 | 4. Bring down 2. 5. Multiply 2 by 2 = 4, add to -3 = 1. 6. Multiply 2 by 1 = 2, add to 4 = 6. 7. Multiply 2 by 6 = 12, add to -5 = 7. Bottom row: $2, 1, 6, 7$. So quotient is $2x^2 + x + 6$ and remainder is 7. Therefore: $$\frac{2x^3 - 3x^2 + 4x - 5}{x - 2} = 2x^2 + x + 6 + \frac{7}{x - 2}$$