Sure, let's find [tex]\( f(c) \)[/tex] using synthetic substitution for the given polynomial [tex]\( f(x) = -x^7 + 4x^6 - 4x^5 - 6x^4 - 3x^3 - 6x^2 - x - 2 \)[/tex] at [tex]\( c = -5 \)[/tex].
Here's a step-by-step method to carry out the synthetic substitution:
1. Set Up the Coefficients:
[tex]\[
\text{Coefficients of } f(x): -1, 4, -4, -6, -3, -6, -1, -2
\][/tex]
2. Write Down the Value of [tex]\( c \)[/tex]:
[tex]\[
c = -5
\][/tex]
3. Perform the Synthetic Substitution Process:
- Write down the coefficient sequence.
- Draw a horizontal line below it.
- Write [tex]\( c \)[/tex] to the left outside the grid.
[tex]\[
\begin{array}{r|rrrrrrrr}
-5 & -1 & 4 & -4 & -6 & -3 & -6 & -1 & -2 \\
\hline
& & & & & & & & \\
& -1 & & & & & & & \\
\end{array}
\][/tex]
4. Carry Down the Leading Coefficient: First step is to carry down the leading coefficient (-1):
[tex]\[
\begin{array}{r|rrrrrrrr}
-5 & -1 & 4 & -4 & -6 & -3 & -6 & -1 & -2 \\
\hline
& & & & & & & & \\
& -1 & & & & & & & \\
\end{array}
\][/tex]
5. Multiply and Add:
- Multiply the carried-down number by [tex]\( c = -5 \)[/tex] and write the result under the next coefficient.
- Add the numbers in the second row and write the result in the third row.
[tex]\[
\begin{array}{r|rrrrrrrr}
-5 & -1 & 4 & -4 & -6 & -3 & -6 & -1 & -2 \\
\hline
& & & & & & & & \\
& -1 & 5 & & & & & & \\
\end{array}
\][/tex]
Now:
[tex]\[
4 + 5 = 9
\][/tex]
Continue this process:
[tex]\[
\begin{array}{r|rrrrrrrr}
-5 & -1 & 4 & -4 & -6 & -3 & -6 & -1 & -2 \\
\hline
& & & & & & & & \\
& -1 & 5 & 9 & -49 & 239 & -1188 & 5934 & -29671 \\
\end{array}
\][/tex]
Check this calculation continuing step-by-step:
[tex]\[
\begin{array}{r|rrrrrrrr}
-5 & -1 & 4 & -4 & -6 & -3 & -6 & -1 & -2 \\
\hline
& & & & & & & & \\
& -1 & 5 & -21 & -94 & 464 & -2312 & 11559 & -57807 \\
-1 & 5 & -21 & 94 & -437 & 2301 & -11695 & 57975 & -288797
\end{array}
\][/tex]
Thus, following all the steps of our synthetic substitution:
[tex]\[
f(-5) = 149,603
\][/tex]
This yields the desired output.
