Remainder theorem: checking factors

so we're asked is the expression X -3 is this a factor of this fourth degree polynomial and you could solve this by doing algebraic long division by taking all of this business and dividing it by X minus 3 and figuring out if you have a remainder if you if you do end up with a remainder then this is not a factor of of this but if you don't have a remainder then that means that this divides fully into this right over here without a remainder which means it is a factor so if the remainder is equal to 0 the remainder is equal to 0 if and only if it's a factor it is a factor and we know a very fast way of calculating the remainder of when you take some polynomial and you divide it by a first-degree expression like this I guess you could say when you divide it by a first-degree polynomial like this the polynomial remainder theorem the polynomial remainder theorem tells us that if we take some let me do this some polynomial P of X and we were to divide it by some X minus a then the remainder is just going to be equal to our polynomial evaluated at our polynomial evaluated a so let's just see what's a in this case well in this case our a is positive 3 so let's just evaluate our polynomial at x equals 3 if what we get is equal to 0 that means our remainder is 0 and that means that X minus 3 is a factor if we get some other remainder that means what we have a nonzero remainder and this isn't a factor so let's try it out so we're going to have so I'm just going to do it I'll do it all in magenta it might be a little computationally intensive so it's going to be 2 times 3 to the fourth power 3 to the 4/3 or Thursday is 81 81 minus 11 yeah this is going to get a little computationally intensive but let's let's see if we can power through it 11 times 27 I probably should have picked a simpler example but let's just keep going plus 15 times 9 plus 4 times 3 is 12 minus 12 so lucky for us at least those last two terms cancel out and this is going to be the rest from here is arithmetic two times 81 is 162 now let's think about what 27 times 11 is so let's see 27 times 10 is going to be 270 270 plus another 27 is minus 297 297 did I do that yeah 270 so 27 times 10 is 270 plus 27 27 yep that's right and then we have I'm prone to make careless errors here see 90 plus 45 is 135 so plus 135 and let's see if I were to take if I were to take 162 and 135 that's going to give me 297 minus 297 minus 200 do that green color minus 297 and we do indeed equal zero so the remainder when I were to divide this by this is equal to 0 so X minus 3 is indeed a factor of all of this business