3 | 1 5 -8 6
. | 3 24 48
- - - - - - - - - - - - - - - -
. | 1 8 16 54
That is to say,
[tex]\dfrac{x^3+5x^2-8x+6}{x-3}=x^2+8x+16+\dfrac{54}{x-3}[/tex]
The remainder is 54.
Another way of doing it is to apply the polynomial remainder theorem, which says the remainder upon dividing a polynomial [tex]p(x)[/tex] by [tex]x-c[/tex] is exactly [tex]p(c)[/tex]. So recognizing that the listed coefficients refer to
[tex]p(x)=x^3+5x^2-8x+6[/tex]
we find
[tex]p(3)=3^3+5\cdot3^2-8\cdot3+6=54[/tex]
