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,

`(x^3+5x^2-8x+6)/(x-3)=x^2+8x+16+(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`