To find [tex]\( f(c) \)[/tex] using synthetic substitution for [tex]\( f(x) = 5x^5 + 10x^4 + 3x^3 + 8x^2 - 6x - 3 \)[/tex] and [tex]\( c = 3 \)[/tex], follow these steps:
1. List the coefficients of the polynomial:
[tex]\[
5, \quad 10, \quad 3, \quad 8, \quad -6, \quad -3
\][/tex]
2. Start with the leading coefficient:
Place [tex]\(5\)[/tex] as the initial value.
[tex]\[
\text{Initial:} \quad 5
\][/tex]
3. Multiply this value by [tex]\(c = 3\)[/tex] and add the next coefficient:
[tex]\[
5 \times 3 + 10 = 15 + 10 = 25
\][/tex]
4. Repeat this process with the new result:
[tex]\[
25 \times 3 + 3 = 75 + 3 = 78
\][/tex]
[tex]\[
78 \times 3 + 8 = 234 + 8 = 242
\][/tex]
[tex]\[
242 \times 3 - 6 = 726 - 6 = 720
\][/tex]
[tex]\[
720 \times 3 - 3 = 2160 - 3 = 2157
\][/tex]
5. List out intermediate results for clarity:
[tex]\[
\begin{align*}
1.\ 5 \\
2.\ 25 \\
3.\ 78 \\
4.\ 242 \\
5.\ 720 \\
6.\ 2157
\end{align*}
\][/tex]
6. The final result [tex]\( f(c) = 2157 \)[/tex]:
Thus, the final value of the polynomial evaluated at [tex]\( c = 3 \)[/tex] is:
[tex]\[
f(3) = 2157
\][/tex]
Interpreting the intermediate results sequence:
[tex]\[
[5, 25, 78, 242, 720, 2157]
\][/tex]
So, the final result from synthetic substitution is:
[tex]\[
f(3) = 2157
\][/tex]
