To determine the missing number in the synthetic division table, let's follow the steps involved in synthetic division. The provided synthetic division table suggests it represents the division of a polynomial by [tex]\( x - 2 \)[/tex].
Here's the synthetic division process in detail:
1. Start with the polynomial coefficients: 2, -2, 3, and 4.
2. We are dividing by [tex]\( x - 2 \)[/tex], so we use the root [tex]\( 2 \)[/tex] for synthetic division.
[tex]\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 3 & 4 \\
& & 4 & 4 & 14 \\ \hline
& 2 & 2 & ? & 18
\end{array}
\][/tex]
To fill in the missing number, follow these steps:
1. Bring down the first coefficient (2) unchanged:
[tex]\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 3 & 4 \\
& & & & \\ \hline
& 2 & & &
\end{array}
\][/tex]
2. Multiply the root (2) by the value just written down (2) and write the result (4) under the next coefficient:
[tex]\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 3 & 4 \\
& & 4 & & \\ \hline
& 2 & & &
\end{array}
\][/tex]
3. Add the second coefficient (-2) to this product (4) to get 2:
[tex]\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 3 & 4 \\
& & 4 & & \\ \hline
& 2 & 2 & &
\end{array}
\][/tex]
4. Multiply the root (2) by the value just written down (2) to get 4, then write this product under the next coefficient:
[tex]\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 3 & 4 \\
& & 4 & 4 & \\ \hline
& 2 & 2 & &
\end{array}
\][/tex]
5. Add the third coefficient (3) to the product (4) to find the missing number, which is 7:
[tex]\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 3 & 4 \\
& & 4 & 4 & \\ \hline
& 2 & 2 & 7 &
\end{array}
\][/tex]
6. Finally, multiply the root (2) by the number just written down (7) to get 14, and add it to the next coefficient (4) to get 18:
[tex]\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 3 & 4 \\
& & 4 & 4 & 14 \\ \hline
& 2 & 2 & 7 & 18
\end{array}
\][/tex]
Thus, the missing number is
[tex]\[ 7 \][/tex]
Therefore, the answer is [tex]\( \boxed{7} \)[/tex].
