Question

GIVING MANY POINTS!

Let p=x^999 − x^100+3x^9 − 5 and q=x + 1. Since q has degree 1, it follows that the remainder when p is divided by q is a constant function k, for some k. What is the value of k?

Answer 1

The polynomial remainder theorem gives an immediate answer. It says that the remainder upon dividing
`p(x)` by
`x-c` is exactly
`p(c)`. In this case
`q=x+1\implies c=-1`, and we have

`k=p(-1)=(-1)^(999)-(-1)^(100)+3(-1)^9-5=-1-1-3-5=-10`