From the new blue book, test #1, section 4, #29:
For a polynomial p(x), the value of p(3) is −2. Which of the following must be true about p(x) ?
A) x − 5 is a factor of p(x).
B) x − 2 is a factor of p(x).
C) x + 2 is a factor of p(x).
D) The remainder when p(x) is divided by x − 3 is −2.

Yeah, this is a tricky one. I actually think the way many students will get it is by elimination. Given p(3) = –2, we know nothing about x – 5, x – 2, or x + 2. So, choice D might not be intuitive, but at least it’s related to p(3) somehow.

Remember that if p(3) = 0, then we know that x – 3 is a factor of p(x). Here’s the tricky bit: if p(3) = –2, then p(3) + 2 = 0. In other words, if we shift p(x) up 2, then we’ll have a polynomial that’s divisible by x – 3. If that’s the case, then the remainder when p(x) is divided by x – 3 must be –2.

Really, the best way to understand this is to see an example. Let’s make up a function: f(x)=(x-2)(x-3). We can FOIL that out: f(x)=x^2-5x+6. When we graph that, we see that there are zeros at x = 2 and x = 3. And because we made the function the way we did, we already know that it’s divisible by x – 2 and x – 3.

function1

Now, what happens when we shift that function down by 2? f(x)-2=x^2-5x+6-2=x^2-5x+4. Let’s call that g(x). Obviously now we don’t have zeros at x = 2 and x = 3. We have –2s instead: g(2) = g(3) = –2.

function2

What happens when we divide g(x) by x – 2?

     \begin{align*} &x-3\\ x-2|&\overline{x^2-5x+4}\\ &\underline{x^2-2x}\\ &\:\:\:\:\:\:{-3x+4}\\ &\:\:\:\:\:\:\underline{-3x+6}\\ &\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:-2 \end{align*}

That’s right—we get x – 3 with a remainder of –2.