 This video will talk about the 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. So 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 in chug negative 2 into this function, we would end up with negative 8. It also asks us to do it with positive 2. 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. 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 16 minus 24. And 16 minus 24 is a negative 8. So it really does work. Now the factor theorem says that 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 said it 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 now 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, 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 1, 0 for the squared term, negative 13 and 12. 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. Yes, it is a 0. This time I'll also show you the other way that you can use the factor theorem. We could take our negative 4 and just plug and chug it. So we're really doing P of negative 4. So that means negative 4 is going to be cubed. And then minus 13 times our negative 4 plus the 12. And when you do that, negative 4 cubed is negative 64. Negative 13 times negative 4 is plus 52 and then plus our 12. 52 plus 12 is 64 minus 64 is going to be 0. You have x equal negative 4 is a 0. So let's look at a story problem. Due to fluctuation in tax revenues, the county government is projecting a deficit in the next 12 months, followed by a quick recovery, payment of all the debt near the end of the period, and we got this function. And it represents the amount of debt in millions of dollars in month M. So use a remainder theorem to find that the debt was higher in month 5 or month 10 and how much higher. Well, remember if we do the remainder theorem, it tells us that if we do our synthetic division here, that whatever we get down here will be what the deficit actually was. I remember P of 5, the outcome is what we would get. So we have 0.1 and negative 2 and 15 and negative 64 and negative 3. So let's bring it down, 0.1. And then 0.1 times 5 would be 0.5. And I've already done all these calculations so you can check them on your own if you want to. But negative 2 plus 0.5 is going to be negative 1.5. That one's not so bad. But negative 1.5 times 5 is going to be negative 7.5. And 15 plus negative 7.5 is going to be 7.5. 7.5 times 5 is going to give us 37.5. And if I add 37.5 to negative 64, I get a negative 26.5. And negative 26.5 times 5 is a negative 132.5. So my remainder or actually the result when I plug it in is negative 135.5. There's a decimal in there. So let's try it with 10. Now let's look at the month 10. Now that took a lot of time to do all that. So remember that we can also do F or in this case D of 10. And to do that I just actually put the formula in my calculator. And I'm going to go set my window up so that I can look at 10. Actually I want to verify 5 so I'll go to 5 first. And I'll do second graph. And at 5 I have negative 135 like I expected. And at 10 I'm going to get negative 143. So if I have negative 143 and I want to find the difference and I have minus a negative 135.5 I end up with a difference of 7.5 when I add the opposite. Or we can do it this way. Go to my home screen and take negative 143 minus a negative 135.5 and I get negative 7.5. And that would be million dollars. So now it says the total debt reaches maximum between month 7 and 10 using remainder theorem to determine which month I've put in this function D of M. And we want to know what happens where the maximum is between 7 and 10. So I'm going to go into my second window and look at it starting at 7. And I'll go every one. So I'm ready to go to second graph to look at my table. And I can see that at 7 I have negative 161.9. At 8 I have negative 169.4. And then I go back down at month 9 to 165.9. So our maximum happened at 8 months.