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Using the Polynomial Remainder Theorem: checking factors


Video transcript

- [Voiceover] So we're asked, Is the expression x minus three, 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 three and figuring out if you have a remainder. If you do end up with a remainder then this is not a factor 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 zero, the remainder is equal to zero, 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 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 at a. So let's just see what's a in this case. Well in this case our a is positive three. So let's just evaluate our polynomial at x equals 3, if what we get is equal to zero that means our remainder is zero and that means that x minus three is a factor. If we get some other remainder that means well we have a non-zero remainder and this isn't a factor, so let's try it out. So, we're gonna have, so I'm just gonna do it all in magenta. It might be a little computationally intensive. So it's going to be two times three to the fourth power, three to the fourth, three to (mumbles), that's 81. 81. Minus 11 Yeah, this is gonna get a little computationally intensive but 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 nine. Plus four times three is 12. Minus 12 So lucky for us, at least those last two terms cancel out. And so 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, 297 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, we do it in that green color, minus 297. And we do indeed equal zero. So the remainder, if I were to divide this by this, is equal to zero. So x minus three is indeed a factor of all of this business.