Let's break down the synthetic division process step-by-step for the polynomial [tex]\( F(x) = x^3 - 2x^2 - 5x + 6 \)[/tex] with an upper bound of 2.
1. Initial Setup:
- Coefficients of [tex]\( F(x) \)[/tex]: [1, -2, -5, 6]
- Upper bound: 2
2. Synthetic Division Table:
- Write down the coefficients: [tex]\( 1, -2, -5, 6 \)[/tex]
3. Bring Down the First Coefficient:
- The first coefficient is always brought down as it is: [tex]\( 1 \)[/tex]
4. Perform Synthetic Division:
- Multiply the upper bound (2) by the value just written down (1) and add it to the next coefficient (-2):
[tex]\[ 2 \times 1 + (-2) = 0 \][/tex]
- Write down the result: [tex]\( 0 \)[/tex]
- Repeat the process: Multiply the upper bound (2) by the last result (0) and add it to the next coefficient (-5):
[tex]\[ 2 \times 0 + (-5) = -5 \][/tex]
- Write down the result: [tex]\( -5 \)[/tex]
- Finally, repeat again: Multiply the upper bound (2) by the last result (-5) and add it to the next coefficient (6):
[tex]\[ 2 \times (-5) + 6 = -4 \][/tex]
- Write down the result: [tex]\( -4 \)[/tex]
The synthetic division process gives us the result:
[tex]\[ [1, 0, -5, -4] \][/tex]
Given this synthetic division result, we can compare to the given options:
Option B matches the synthetic division table and has the desired result:
\begin{tabular}{rrrr}
& 2 & 0 & -10 \\
\hline
1 & 0 & -5 & -4
\end{tabular}
So, option B represents the correct synthetic division result.
