Use synthetic division to find
the quotient.

Ask by Lane Warren. in the United States

Jan 21,2025

Question

Use synthetic division to find the quotient. $$ \frac{2 x^{3}+8 x+13}{x-2} $$ $$ [?] x^{2}+\square x+\square+\frac{\square}{x-2} $$

UpStudy AI · Accepted Answer

To perform synthetic division for the polynomial $$ \frac{2x^{3} + 8x + 13}{x - 2} $$, follow these steps:

1. **Set Up the Synthetic Division:**
   - **Divisor Root:** Since we are dividing by $$ x - 2 $$, the root is $$ 2 $$.
   - **Polynomial Coefficients:** Write down the coefficients of the dividend. Note that the polynomial $$ 2x^3 + 8x + 13 $$ is missing the $$ x^2 $$ term, so include a $$ 0 $$ for it. Thus, the coefficients are $$ 2, 0, 8, 13 $$.

2. **Perform the Synthetic Division:**

```
   2 |  2    0     8     13
          4     8     16
        ----------------------
           2    4    16     29
   ```

- **Step-by-Step Calculation:**
     - **Bring down the first coefficient:** $$ 2 $$.
     - **Multiply by the root (2):** $$ 2 \times 2 = 4 $$.
     - **Add to the next coefficient:** $$ 0 + 4 = 4 $$.
     - **Multiply by the root (2):** $$ 4 \times 2 = 8 $$.
     - **Add to the next coefficient:** $$ 8 + 8 = 16 $$.
     - **Multiply by the root (2):** $$ 16 \times 2 = 32 $$.
     - **Add to the last coefficient:** $$ 13 + 32 = 45 $$.

3. **Interpret the Result:**
   - The numbers on the bottom row represent the coefficients of the quotient and the remainder.
   - The quotient is $$ 2x^2 + 4x + 16 $$.
   - The remainder is $$ 45 $$.

4. **Write the Final Answer:**
   $$
   \frac{2x^{3} + 8x + 13}{x - 2} = 2x^{2} + 4x + 16 + \frac{45}{x - 2}
   $$

**Filled in the blanks:**
$$
\boxed{2\, x^{2}+4\, x+16+\frac{45}{x-2}}
$$
The quotient is $$ 2x^{2} + 4x + 16 $$ with a remainder of $$ 45 $$, so the expression becomes $$ 2x^{2} + 4x + 16 + \frac{45}{x - 2} $$.

Use synthetic division to find
the quotient.

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