Sure, let's tackle this synthetic-division problem step-by-step!
Given the polynomial [tex]\( 2x^4 + x^3 - 2x^2 - 8x + 5 \)[/tex] and dividing it by [tex]\( x - 2 \)[/tex]:
1. Set up the synthetic division: Write down the coefficients of the polynomial:
[tex]\[
\begin{array}{ccccc}
2 & 1 & -2 & -8 & 5 \\
\end{array}
\][/tex]
2. Divide by [tex]\( x - 2 \)[/tex]: The divisor here is 2.
3. Carry down the leading coefficient (which is 2) to the bottom row:
[tex]\[
\begin{array}{ccccc}
& 2 & 1 & -2 & -8 & 5 \\
2 & & & & & 2 \\
\end{array}
\][/tex]
4. Multiply the 2 (bottom row) by the divisor (2) and write the result underneath the next coefficient (1):
[tex]\[
\begin{array}{ccccc}
& 2 & 1 & -2 & -8 & 5 \\
2: & - & 4 & & & 2 \\
\end{array}
\][/tex]
Here, [tex]\( 2 \times 2 = 4 \)[/tex].
5. Add the numbers in the second column: [tex]\( 1 + 4 = 5 \)[/tex]:
[tex]\[
\begin{array}{ccccc}
& 2 & 1 & -2 & -8 & 5 \\
2: & & 4 & & & 2 \\
& 2 & 5 & & & 2 \\
\end{array}
\][/tex]
6. Multiply the recent bottom row value by the divisor (2) and place the result in the next column:
[tex]\[
\begin{array}{ccccc}
& 2 & 1 & -2 & -8 & 5 \\
2: & & 4 & 10 & & 2 \\
& 2 & 5 & 8 & & 2 \\
\end{array}
\][/tex]
Here, [tex]\( 5 \times 2 = 10 \)[/tex].
7. Add the numbers in the third column: [tex]\( -2 + 10 = 8 \)[/tex]:
[tex]\[
\begin{array}{ccccc}
& 2 & 1 & -2 & -8 & 5 \\
2: & & 4 & 10 & 16 & 2 \\
& 2 & 5 & 8 & 8 & 2 \\
\end{array}
\][/tex]
8. Next, [tex]\( 8 \times 2 = 16 \)[/tex] (place in the next column):
[tex]\[
\begin{array}{ccccc}
& 2 & 1 & -2 & -8 & 5 \\
2: & & 4 & 10 & 16 & & 2 \\
& 2 & 5 & 8 & 8 & 21 & 2 \\
\end{array}
\][/tex]
9. Add the numbers in the fourth column: [tex]\( -8 + 16 = 8 \)[/tex]:
[tex]\[
\begin{array}{ccccc}
& 2 & 1 & -2 & -8 & 5 \\
2: & & 4 & 10 & 16 & & 2 \\
& 2 & 5 & 8 & 16 & 2 \\
\end{array}
\][/tex]
10. Finally, multiply [tex]\( 8 \times 2 = 16 \)[/tex] (place in the next column):
[tex]\[
\begin{array}{ccccc}
& 2 & 1 & -2 & -8 & 5 \\
2: & & 4 & 10 & 16 & 21 & 2 \\
\end{array}
\][/tex]
11. Add the numbers in the fifth column: [tex]\( 5 + 16 = 21 \)[/tex].
The final coefficients after synthetic division are:
[tex]\[
2, 5, 8, 8, 21
\][/tex]
So, the numbers obtained from the synthetic division are [tex]\(2, 5, 8, 8, 21\)[/tex].
Thus, the number that belongs in the place of the question mark is [tex]\(21\)[/tex].
