Question

What’s the remainder for this synthetic division problem?

Answer 1

3   |   1    5     -8     6

.    |         3    24   48

- - - - - - - - - - - - - - - -

.    |    1    8    16    54

That is to say,

$\dfrac{x^3+5x^2-8x+6}{x-3}=x^2+8x+16+\dfrac{54}{x-3}$

The remainder is 54.

Another way of doing it is to apply the polynomial remainder theorem, which says the remainder upon dividing a polynomial $p(x)$ by $x-c$ is exactly $p(c)$. So recognizing that the listed coefficients refer to

$p(x)=x^3+5x^2-8x+6$

we find

$p(3)=3^3+5\cdot3^2-8\cdot3+6=54$