At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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].
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].
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
