Remainder theorem examples

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 3 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 equals 0 or just what would happen if I evaluate 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 know you would 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 it 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 2 has a remainder of 1 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 2 has a remainder of 1 that tells us that P of not of negative 2 but P of positive 2 whatever x value would make this expression equal 0 that P of 2 is equal to 1 and then we could use this top information to figure out what P of 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 eleven 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 2 to the fourth again so it's minus 16 plus 4 K minus 11 is equal to 1 and these cancel out and let's see we can add 11 to both sides of this equation and we get 4 K 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 2 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 alright 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 P of positive 3 is equal to 5 and similarly what is the remainder actually no not so similar this is interesting what does remainder when P of X is divided by X I know what your thinking is like wait what 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 let me be like oh now I get it or if I wrote divided by X minus 0 like oh now I get it this is going to be P of and it doesn't matter either 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