Question

Use synthetic division to determine if the given value for kk is a zero of this polynomial. If not, determine p( k )p( k ).
p(x) = 3x3 − 6x2 − 8x − 19; k = 3

Answer 1

3 | 3 ... -6 ... -8 ... -19
.. | .. ...  9  ..  9 ....    3
- - - - - - - - - - - - - - - -
.. | 3 ...  3 ...  1 ....  -16

which means

$\dfrac{3x^3-6x^2-8x-19}{x-3}=3x^2+3x+1-\dfrac{16}{x-3}$

which is to say that $p(x)$ is not divisible by $x-3$, and so $x=3$ is not a root. This in turn means $p(k)\neq0$.

Indeed, you have

$p(3)=3(3)^3-6(3)^2-8(3)-19=81-54-24-19=-16$

If you were familiar with the polynomial remainder theorem, you would have known this immediately from determining the remainder alone.