Answer
$p(0) = -4$
$p(1) = -3$
$p(-3) = 101$
$p(7) = 5001$
Work Step by Step
We are given the polynomial $p(x) = 2x^4 + x^3 - 3x^2 + x - 4$ and asked to find $p(0), p(1), p(-3),$ and $p(7)$ using synthetic division and the Remainder Theorem. We can directly evaluate the polynomial at each point.
Find $p(0)$:
We divide $p(x)$ by $x$:
\[
\begin{array}{r|rrrr}
& 2 & 1 & -3 &1 &-4 \\ \hline
0 & & 0 & 0 & 0 & 0 \\ \hline
& 2 & 1 & -3 & -1 & -4\\
\end{array}
\] The rest is $-4$, so $p(0)=-4$.
Find $p(1)$:
We divide $p(x)$ by $x-1$:
\[
\begin{array}{r|rrrr}
& 2 & 1 & -3 &1 &-4 \\ \hline
1 & & 2 & 3 & 0 & 1 \\ \hline
& 2 & 3 & 0 & 1 & -3\\
\end{array}
\] The rest is $-3$, so $p(1)=-3$.
Find $p(-3)$:
We divide $p(x)$ by $x+3$:
\[
\begin{array}{r|rrrr}
& 2 & 1 & -3 &1 &-4 \\ \hline
-3 & & -6 & 15 & -36 & 105 \\ \hline
& 2 & -5 & 12 & -35 & 101\\
\end{array}
\] The rest is $101$, so $p(-3)=101$.
Find $p(7)$:
We divide $p(x)$ by $x-7$:
\[
\begin{array}{r|rrrr}
& 2 & 1 & -3 &1 &-4 \\ \hline
7 & & 14 & 105 & 714 & 5005 \\ \hline
& 2 & 15 & 102 & 715 & 5001\\
\end{array}
\] The rest is $5001$, so $p(7)=5001$.
