Remainder theorem examples | Polynomial Division | Algebra 2 | Khan Academy

So we have the graph here of y is equal to p of x. I could write it like this: y is equal to p of x. And they say, what is the remainder when p of x is divided by x plus three? So pause this video and see if you can have a go at this. And they tell us your answer should be an integer.

So, as you might have assumed, this will involve the polynomial remainder theorem. And all that tells us is that, hey, if we were to take p of x and divide it by x plus 3, whatever the remainder is here, so let's say the remainder is equal to k, that value k is what we would have gotten if we took our polynomial and we evaluated it at the value of x that would have made x plus 3 equal 0. Or just, what would happen if I evaluated our polynomial at x equals negative 3?

You have to be very careful there. Sometimes people get confused; they see a positive 3 and then they evaluate the polynomial at the positive 3 to figure out the remainder. No, you would. If you saw a positive 3 there, you would evaluate the polynomial at negative 3. But this should be equal to k as well.

And so what is the remainder when p of x is divided by x plus 3? Well, it's going to be equal to p of negative 3. p of negative 3 looks like it is equal to negative 2. It is equal to negative 2. So our remainder is equal to negative 2 in this situation.

Let's do another example. Actually, let's do several more examples here. We're told that p of x is equal to all of this business where k is an unknown integer. Very interesting! p of x divided by x minus two has a remainder of one. What is the value of k? So pause this video again; see if you can work it out.

All right, well, this second sentence—that p of x divided by x minus two has a remainder of one—that tells us that p of, not of negative two, but p of positive two, whatever x value would make this expression equal zero, that p of 2 is equal to 1.

And then we could use this top information to figure out what p of 2 would be. It would be 2 to the 4th power minus 2 times 2 to the third power plus k times 2 squared, so times 2 squared minus 11. And so all of that, that's p of 2 right over here, that's going to be equal to 1.

2 to the fourth is 16, and then 2 times 2 to the third—that's 2 to the 4th again—so it's minus 16 plus 4k minus 11 is equal to 1. These cancel out. And let's see, we can add 11 to both sides of this equation, and we get 4k is equal to 12. Divide both sides by 4, and we get k is equal to 3, and we're done.

Let's do another example. In fact, let's do two more because we're having so much fun. So this next question tells us p of x is a polynomial, and they tell us what p of x divided by various things are—what the remainder would be when you divide p of x by these various expressions. Find the following values of p of x: p of negative 4 and p of 1. Pause this video and see if you can have a go at it.

All right, so p of negative 4—this is going to be equal to the remainder. Remainder when p of x divided by what you might be tempted to say x minus 4, but they're trying to trick you intentionally. This would be the remainder when p of x is divided by x plus 4.

And so they tell us right over here p of x divided by x plus 4 has a remainder of 3. So it's going to be 3 right over there. And similarly, p of 1—this is going to be the remainder. This is the remainder when p of x divided by not x plus 1, but x minus 1. So when p of x is divided by x minus 1, the remainder is 0.

Let's do one last example. So once again, p of x is a polynomial, and then they give us a few values of p of x, and they say, what is the remainder when p of x is divided by x minus 3? Pause the video and try to think about that.

Well, we've gone over this multiple times. The remainder when p of x is divided by x minus 3, that would be p of not negative 3, p of positive 3—whatever value makes, whatever value of x makes this entire expression equal 0. So p of positive 3 is equal to 5.

And similarly, what is the remainder? Actually, no, not so similar. This is interesting. What is the remainder when p of x is divided by x? I know what you're thinking. It's like, wait what? What number am I dealing with? But if I were to rewrite this, instead of saying divided by x, if I were to say divided by x plus 0, then you'd be like, oh, now I get it.

Or if I wrote divided by x minus 0, you're like, oh, now I get it. This is going to be p of—and it doesn't matter whether I take a positive or a negative 0—it's going to be p of 0. And p of 0 they tell us is negative 1, and we're done.