Will mark ! given p(x)=x3−4x2+15x+k and the remainder of the division of p(x) by (x+1) is equal to −8, what is the remainder of the division of p(x) by (x−1)? −8 8 12 24

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 .