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------------------------------------------------ What is the remainder in the synthetic division problem below?

[tex]\[

-2 \quad \begin{array}{|cccc}

1 & 2 & -3 & 1

\end{array}

\][/tex]

A. 13
B. 9
C. 11
D. 7

Sure! Let's solve the synthetic division problem step-by-step:

We're given a polynomial with coefficients [tex]$1, 2, -3, 1$[/tex] and we need to divide it by [tex]$x + 2$[/tex]. In synthetic division, using [tex]$x + 2$[/tex] means our divisor value, or root, is [tex]$-2$[/tex].

Here’s how we solve it:

1. Set up the synthetic division: Write the coefficients of the polynomial in a row: [tex]$1, 2, -3, 1$[/tex].

2. Write the divisor (root): This is [tex]$-2$[/tex].

3. Start with the left-most coefficient: Bring down the first coefficient ([tex]$1$[/tex]) as it is.

4. Multiply and add: Follow these steps for each column:
- Multiply the current number at the bottom by the divisor ([tex]$-2$[/tex]).
- Write the result under the next coefficient.
- Add this result to the next coefficient and write the sum below the line.

Here is the step-by-step breakdown:

- First column:
- Bring down [tex]$1$[/tex].

- Second column:
- Multiply [tex]$1 \times (-2) = -2$[/tex].
- Add to the next coefficient: [tex]$2 + (-2) = 0$[/tex].

- Third column:
- Multiply [tex]$0 \times (-2) = 0$[/tex].
- Add to the next coefficient: [tex]$-3 + 0 = -3$[/tex].

- Fourth column:
- Multiply [tex]$-3 \times (-2) = 6$[/tex].
- Add to the next coefficient: [tex]$1 + 6 = 7$[/tex].

5. Remainder: The last number you get is the remainder of the division. So, the remainder here is [tex]$7$[/tex].

The remainder in the synthetic division problem is [tex]$7$[/tex]. So, the correct answer is option D. 7.