Remainder theorem gives negative results?

I always feel pretty dense when posting here, especially given how complicated everyone else's problems seem! (But hey, I guess that works in inverse, being able to answer my questions because they are so ridiculously easy, making you feel smart!)

So, I'm studying using the past exam papers, and come across this question.

f(x)=3x^3-5x^2-16x+12.
Find the remainder when f(x) is divided by (x-2).
f(2)=(3×8)-(5×4)-(16×2)+12.
24-20-32+12=-16.

I don't see the logistical sense in this problem. You can have a negative remainder? Have I multiplied something out wrong or is it right? I don't know, but it doesn't look right and I can't find it in my revision guide. It just explains that something is a factor when the remainder is -, proving to be unhelpful.

Reply 1

DFranklin

Yes, you can have a negative remainder.

It isn't true that something is a factor when the remainder is -ve, however. It's only a factor if the remainder is zero.

Reply 2

2^1/2

nothing wrong with -ve remainders to my knowlege

Reply 3

Lanares

So, it's just me being dense then.

(I didn't mean to type -, I meant to type 0! That's what it says in the book, lol. Typo.)

I was absent for this lesson and had to teach it to myself, missing all of his input and clever tips. I guess I just didn't understand that you could have a negative remainder because the idea of a remainder is to see what remains. If less than nothing remains -, grarasrh! I'm mind-boggled. Haha. Thank you all!

Reply 4

Swayum

You can't really think of it in the same way as dividing 9 by 5 and getting a remainder of 4. Try dividing f(x) by x-2 without using the remainder theorem (in other words, actually carry out the long division). You'll see that you end up with -16 at the end. You can't do anything with this -16 and so it's the bit "left over" when doing the division, and so, in other words, it's the remainder.