Finding the remainder of an algebraic quotient

[snoopies622]

I'm tutoring a pupil for a CLEP exam and her book includes the following algebra problem:
What is the remainder when
$$9x^{23} - 7x^{12} - 2x^{5} +1$$
is divided by $x+1$?
I know how to find the answer by computing the quotient of these two expressions, but in this case doing that is so tedious I assume there's a more direct way of finding the remainder. What is it?

Edit : I think this might be more appropriately placed in the "Homework and Coursework" section.

 

[zoki85]

Ruffini's rule speeds up the proces when divisior is in form "x-r"

 

[snoopies622]

Indeed it does, thanks zoki85!

 

[HallsofIvy]

Since the problem is only to find the remainder, even simpler is "The remainder when dividing polynomial function P(x) by x- r is P(r)".
That's easy to prove: let Q(x) be the quotient when P(x) is divided by x- r. The P(x)= Q(x)(x- r)+ remainder. Letting x= r give P(r)= Q(r)(0)+ remainder or "remainder= P(r)". To find the remainder when $P(x)= 9x^{23}- 7x^{12}- 2x^5+ 1$ is divided by x+1= x- (-1), just calculate
$$P(-1)= 9(-1)^{23}- 7(-1)^{12}- 2(-1)^5+ 1$$.

 

[snoopies622]

Wow! (high fives HallsofIvy)