Sure, let's use synthetic division and the Remainder Theorem to evaluate [tex]\( P(c) \)[/tex] for [tex]\( P(x) = 2x^2 + 13x + 3 \)[/tex] and [tex]\( c = -4 \)[/tex].
### Steps for Synthetic Division:
1. Write down the coefficients of the polynomial [tex]\( P(x) = 2x^2 + 13x + 3 \)[/tex]. These coefficients are:
[tex]\[
2, \quad 13, \quad 3
\][/tex]
2. Set up the synthetic division with [tex]\( c = -4 \)[/tex]. The first coefficient (2) is brought down as is.
[tex]\[
\begin{array}{r|rrr}
-4 & 2 & 13 & 3 \\
& & & \\
\hline
& 2 & &
\end{array}
\][/tex]
3. Multiply and add. Multiply the number that you just brought down by [tex]\( c \)[/tex] and write the result under the next coefficient. Then add the numbers in the column.
[tex]\[
\begin{array}{r|rrr}
-4 & 2 & 13 & 3 \\
& & -8 & \\
\hline
& 2 & 5 &
\end{array}
\][/tex]
4. Repeat the process: Continue multiplying and adding.
[tex]\[
\begin{array}{r|rrr}
-4 & 2 & 13 & 3 \\
& & -8 & -20\\
\hline
& 2 & 5 & -17
\end{array}
\][/tex]
5. The last number in the bottom row is the remainder and it gives [tex]\( P(-4) \)[/tex] for the polynomial [tex]\( P(x) \)[/tex].
Thus, using the remainder theorem, we find:
[tex]\[
P(-4) = -17
\][/tex]
So, [tex]\( P(-4) = -17 \)[/tex].
