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Current time:0:00Total duration:3:39

Remainder theorem: finding coefficients

Video transcript

so for what value of C or values of C is X minus 5 is X Miami write it this way for what for what C is X minus 5 a factor factor of of P of X and I encourage you to pause the video and try to work through it tonight so P is actually a lowercase P P of X all right so I'm assuming you have attempted this and we just have to realize okay if X minus 5 is a factor of P of X that means you could write x ma you could write P of X let me do this in that green color that means you could write P of X as being equal to X minus 5 X minus 5 times some other business times some other polynomial so x times some other polynomial I don't know let's just call it G of X maybe G of X and this would be what would be a second-degree polynomial and so what would P of 5 have to be well P of 5 would have to be 0 5 would have to have be a root of this polynomial you see it right over here if you replace if you replace this with a 5 then this is going to be a 5 and this is going to be a 5 it doesn't matter what G of 5 is this is going to be 0 so if F X at X minus 5 X minus 5 is is a factor is a factor if and only if if and only if P of 5 is equal to 0 P of 5 is equal to 0 you could say that 5 is a root of the polynomial P P of 5 is equal to 0 so let's just set P of 5 equal to 0 and then try to solve for C all right so we're going to get 5 to the third power so this right over here 5 to the third so P of 5 P let me just write down P of 5 is equal to 5 to the third is 125 plus 2 times 5 squared so two times 25 plus 50 plus C times five plus five c plus ten that needs to be equal to zero now let's see if we add if we were to add 125 250 210 we are going to get 175 185 so we get 185 plus 5 C plus 5 C is equal to zero subtract 185 from both sides you get 5 C is equal to negative 185 negative 185 or C C is equal to negative 185 over 5 which is going to be its negative 185 over 5 which is equal to let's see 5 goes into 180 let's see it goes into us 5 let's see it's go be 35 times 30 is 150 5 times 30 is 150 and then we'll have another 35 to go so it's going to be 37 negative 37 is that right 5 times 30 is 150 5 times 7 is 35 yep negative 37 and we're done if this was X to the 3rd plus 2x squared minus 37 X plus 10 then X minus 5 would be a factor of this polynomial