[Math] find variable from polynomial when divisor and remainder is known

Question

I am asked the following question:

When ${x^4 – 3x^3 + px – 5}$ is divided by ${x – 3}$ the remainder is $16$, find $p$.

So my way of solving this was to use synthetic long division to divide ${x^4 – 3x^3 + px – 5}$ by $3$ that would leave me with a remainder expression that would have $p$.
$$
\begin{matrix}3&1&0&-3&p&-5\\
&&3&9&18&54+3p\\
&1&3&6&18+p&49+3p
\end{matrix}
%3| 1 | 0 | -3 | p | -5
% | 3 | 9 | 18 | 54 + 3p
%
% | 1 | 3 | 6 | 18 + p | 49 + 3p
$$

I get the remainder to be ${3p + 49}$

I would say then that ${3p + 49 = 16}$ and ${p = – 11}

But the textbook gives an answer of $7$.

Have I come to the wrong conclusion about how to work out ${p}$?

Answer 1

Hint: Note that $f(x)=(x-3)\cdot g(x)+r(x)$ implies $f(3)=r(3)$.