Remainder theorem examples

- [Instructor] 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 three, whatever the remainder is here, so we'll 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 three equal zero or just what would have happened if I evaluated our polynomial as x equals negative three. You have to be very careful there sometimes people get confused. They see a positive three and then they evaluate the polynomial at the positive three to figure out the remainder. No! If you saw a positive three there, you would evaluate the polynomial at negative three but this should be equal to k as well. And so what is the remainder when P of x is divided by x plus three? Well it's going to be equal to P of negative three. P of negative three it looks like it is equal to negative two. It is equal to negative two, so our remainder is equal to negative two 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 not of negative two but P of positive two, whatever x value would make this expression equal zero. That P of two is equal to one. And then we could use this top information to figure out what P of two would be. It would be two to the fourth power minus two times two to the third power plus k times two squared. So, times two squared minus 11, and so all of that, that's P of two right over here that's going to be equal to one. Two to the fourth is 16 and then two times two to the third that's two to the fourth again, so it's minus 16 plus four k minus 11 is equal to one, these cancel out. Now let's see we can add 11 to both sides of this equation and we get four k is equal to 12. Divide both sides by four and we get k is equal to three 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 divide 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 four and P of one. Pause this video and see if you can have a go at it. All right, so P of negative four, this is going to be equal to the remainder when P of x divided by what. You might be tempted to say x minus four but they're trying to trick you intentionally. This would be the remainder when P of x is divided by x plus four. And so they tell us right over here P of x divided by x plus four has a remainder of three. So, it's going to be three right over there. And similarly P of one, this is going to be the remainder, this is the remainder when P of x divided by not x plus one, but x minus one. So when P of x is divided by x minus one, the remainder is zero. 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 three? 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 three that would be P of not negative three, P of positive three. Whatever value of x makes this entire expression equal zero. So P of positive three is equal to five. And similarly what is the remainder, actually not so similarly, 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 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 zero then you'd be like oh now I get it. Or if I wrote divided by x minus zero, you'd be 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 zero, it's going to be P of zero. And P of zero, they tell us, is negative one and we're done.