Question

Find the quotient and remainder using synthetic divison for : x^3+6x^2+9x+7 / x+2
The quotient is
The remainder is

Answer 1

To perform synthetic division for the polynomial \( x^3 + 6x^2 + 9x + 7 \) divided by \( x + 2 \), we need to use the zero of the divisor, which is \( -2 \).

Here's how to perform the synthetic division:

1. Write down the coefficients of the polynomial \( x^3 + 6x^2 + 9x + 7 \):
- Coefficients: \( 1, 6, 9, 7 \)

2. Set up synthetic division using \( -2 \):

```
-2 | 1 6 9 7
| -2 -8 -2
_____________________
1 4 1 5
```

3. Perform the synthetic division step-by-step:
- Bring down the leading coefficient \( 1 \).
- Multiply \( 1 \) by \( -2 \) and place the result under the next coefficient \( 6 \): \( -2 \).
- Add \( 6 + (-2) = 4 \).
- Multiply \( 4 \) by \( -2 \) and place the result under the next coefficient \( 9 \): \( -8 \).
- Add \( 9 + (-8) = 1 \).
- Multiply \( 1 \) by \( -2 \) and place the result under the next coefficient \( 7 \): \( -2 \).
- Add \( 7 + (-2) = 5 \).

The final row after synthetic division gives the coefficients of the quotient and the remainder:
- The quotient is \( 1x^2 + 4x + 1 \) or simply \( x^2 + 4x + 1 \).
- The remainder is \( 5 \).

Thus, we conclude:

- The quotient is \( x^2 + 4x + 1 \).
- The remainder is \( 5 \).