Absolutely, I can assist with that!
To find [tex]\( f(c) \)[/tex] using synthetic substitution, we'll apply the process to the polynomial [tex]\( f(x) = -x^7 + 4x^6 - 4x^5 - 6x^4 - 3x^3 - 6x^2 - x - 2 \)[/tex] for [tex]\( c = -5 \)[/tex].
Here is the step-by-step synthetic substitution process:
1. Write down the coefficients of the polynomial:
- Coefficients: [tex]\([-1, 4, -4, -6, -3, -6, -1, -2]\)[/tex]
2. Set up the synthetic substitution:
- We have [tex]\( c = -5 \)[/tex].
- Place the coefficients in the synthetic substitution setup.
3. Perform the synthetic substitution:
- Start with the leading coefficient:
[tex]\[
-1
\][/tex]
- Multiply by [tex]\( c \)[/tex] and add the next coefficient:
[tex]\[
-1 \cdot (-5) + 4 = 9
\][/tex]
- Continue this process through all coefficients:
Here is the step-by-step computation:
[tex]\[
\begin{array}{c|rrrrrrrr}
-5 & -1 & 4 & -4 & -6 & -3 & -6 & -1 & -2 \\
\hline
& -1 & 9 & -49 & 239 & -1198 & 5984 & -29921 & 149603 \\
\end{array}
\][/tex]
Breaking it down:
- First step:
[tex]\( -1 \cdot (-5) + 4 \Rightarrow 5 + 4 = 9 \)[/tex]
[tex]\(\rightarrow\)[/tex] Second coefficient becomes 9
- Second step:
[tex]\( 9 \cdot (-5) + (-4) \Rightarrow -45 - 4 = -49 \)[/tex]
[tex]\(\rightarrow\)[/tex] Third coefficient becomes -49
- Third step:
[tex]\( -49 \cdot (-5) + (-6) \Rightarrow 245 - 6 = 239 \)[/tex]
[tex]\(\rightarrow\)[/tex] Fourth coefficient becomes 239
- Fourth step:
[tex]\( 239 \cdot (-5) + (-3) \Rightarrow -1195 - 3 = -1198 \)[/tex]
[tex]\(\rightarrow\)[/tex] Fifth coefficient becomes -1198
- Fifth step:
[tex]\( -1198 \cdot (-5) + (-6) \Rightarrow 5990 - 6 = 5984 \)[/tex]
[tex]\(\rightarrow\)[/tex] Sixth coefficient becomes 5984
- Sixth step:
[tex]\( 5984 \cdot (-5) + (-1) \Rightarrow -29920 - 1 = -29921 \)[/tex]
[tex]\(\rightarrow\)[/tex] Seventh coefficient becomes -29921
- Seventh step:
[tex]\( -29921 \cdot (-5) + (-2) \Rightarrow 149605 - 2 = 149603 \)[/tex]
[tex]\(\rightarrow\)[/tex] Final outcome becomes 149603
Final numbers obtained from the synthetic substitution process are:
[tex]\[
[-1, 9, -49, 239, -1198, 5984, -29921, 149603]
\][/tex]
Thus, [tex]\( f(-5) \)[/tex] evaluates to [tex]\( 149603 \)[/tex].
