2.4.27
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Use the polynomial remainder theorem. If 
is a polynomial of degree 
, then we can divide by a linear term 
to get a quotient 
and remainder 
of the form
Then when
, we get 
. In other words, the value of at tells you the value of the remainder upon dividing by .
So given that
, and the remainder upon dividing by 
is -8, we know that 
, so
Since is a polynomial (not a rational expression), then we know that divides 
exactly. In particular, the remainder term of this quotient is 0. We can use long or synthetic division to determine . I prefer typing out the work for synthetic division:
-1 | 1 -4 15 k + 8
. | -1 5 -20
- - - - - - - - - - - - - - - - - -
. | 1 -5 20 k - 12
The remainder here has to be 0, so
.
Finally, we can get the remainder upon dividing by 
by evaluating 
, which gives 
.
Then when
So given that
Since
-1 | 1 -4 15 k + 8
. | -1 5 -20
- - - - - - - - - - - - - - - - - -
. | 1 -5 20 k - 12
The remainder here has to be 0, so
Finally, we can get the remainder upon dividing