 This video will talk about remainder and factor theorems. The remainder theorem says if a polynomial p of x is divided by x minus c, then using synthetic division, the remainder is equal to, not equation, equal to p of c, or the outcome when we plug in the c. So p of negative 2, we know that c is negative 2, that's on the outside, and then we're going to put in our 1 for x to the fourth and 3 for x cubed, 0 because there's no x squared term, negative 2 for x and then negative 4. Whatever my remainder is here would be the same thing as if I were to plug in this negative 2. So let's do the synthetic division. Bring down 1, multiply by negative 2, and then add this row so we get 1, and 1 times negative 2 would be negative 2 plus 0 would be negative 2. Remember, multiply on the diagonal so that would be positive 4, and then add the column so that gets us positive 2, and 2 times negative 2 would be a negative 4, and that would give us negative 8. So what this is really saying is if we were to plug and chug negative 2 into this function, we would end up with negative 8. It also asks us to do it with positive 2. So again, that means 2 is on the outside, 1, 3, 0, negative 2, negative 4, and bring down the 1 and multiply, add we get 5, 5 times 2 is 10, plus 0 is 10, 10 times 2 is 20, minus 2 is 18, and 18 times 2 is 36, and 36 minus 4 should be 32. And let's just double check one of them. It doesn't matter. So let's do this first one. So we have P of negative 2, and that says that we have negative 2 to the fourth, or that's going to be 16, plus 3 times negative 2 cubed, which is really plus 3 times negative 8, and then minus 2 times negative 2, which we can say is plus 4, and then minus 4. So these two cancel each other out. We really just have negative 8. So it really does work. Now we talk about the factor theorem. If we have the same polynomial, P of x, then if we can find P of c equal to 0, then that would mean that x minus c would become a factor. If you know what c is, you could make it x minus that c, and you've got a factor. Or if you know that x minus c is a factor, then when you plug in the c value into your polynomial, you should get a remainder of 0. So we have here that we want to see if 3x minus 1 is a factor of this. So let's first find out what we have for c. So 3x minus 1 is equal to 0. 3x is going to be equal to 1, because we're really finding the 0 is another way to think about it. And dividing by 3, x is equal to 1 third. That should make things interesting. So 1 third goes on the outside. 3, negative 19, positive 30, negative 8. And if this is really a factor, this should be a 0 down here. But let's multiply. Bring down the 3, multiply by 1 third. 1 third times 3 is just 1. And then we're going to 19 plus 1 is negative 18. And then negative 18 times 1 over 3, you could think of it as negative 18 over 3. If you haven't done fractions for a while. Negative 18 over 3 is negative 6. And negative 6 plus 30 would be a positive 24. And 24 is basically 24 over 3 is going to give me 8. Positive 8 here, remainder 0. So we got what we thought. Okay, now we want to use this factor theorem again to show that the given value is a 0. So we should be able to use our synthetic division again and say negative 4. And then we have bring down the 1, multiply. We get negative 4 and add in 0 and negative 4. We get negative 4. Negative 4 times negative 4 would be positive 16. And when we add, we get 3. And 3 times negative 4 is going to be negative 12. And 12 minus 12 is going to give me 0. So yes, it is a 0. And the factor would be x plus 4. Remember, you bring the number to the other side. And now we've got this nice little factor, which may be actually a little ahead of ourselves.