Divide $$ x^{3}+2 x^{2}-5 x+12 $$ by $$ (x+3) $$

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UpStudy AI · Accepted Answer

To divide $$ x^{3} + 2x^{2} - 5x + 12 $$ by $$ (x + 3) $$, we can use **synthetic division**.

### Steps:

1. **Identify the root**: Since we're dividing by $$ (x + 3) $$, the root is $$ x = -3 $$.

2. **Set up the coefficients** of the polynomial: $$ 1 $$ (for $$ x^3 $$), $$ 2 $$ (for $$ x^2 $$), $$ -5 $$ (for $$ x $$), and $$ 12 $$ (constant term).

3. **Perform synthetic division**:

```
   -3 |  1   2   -5   12
         -3    3    6
         ----------------
           1  -1   -2   18
   ```

- Bring down the first coefficient: **1**.
   - Multiply by **-3** and add to the next coefficient: $$ 1 \times (-3) + 2 = -1 $$.
   - Multiply by **-3** and add to the next coefficient: $$ -1 \times (-3) + (-5) = -2 $$.
   - Multiply by **-3** and add to the constant term: $$ -2 \times (-3) + 12 = 18 $$.

4. **Interpret the result**:
   - The numbers at the bottom represent the coefficients of the quotient and the remainder.
   - The quotient is $$ x^2 - x - 2 $$.
   - The remainder is **18**.

### Final Answer:
$$
\frac{x^{3} + 2x^{2} - 5x + 12}{x + 3} = x^{2} - x - 2 + \frac{18}{x + 3}
$$

Alternatively, expressed with the remainder:
$$
x^{3} + 2x^{2} - 5x + 12 = (x + 3)(x^{2} - x - 2) + 18
$$
The division of $$ x^{3} + 2x^{2} - 5x + 12 $$ by $$ (x + 3) $$ results in $$ x^{2} - x - 2 $$ with a remainder of 18.

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