Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. The remainder and factor theorem for polynomials. If the polynomial F of X = d of X * Q of X + r of X, then we know that the dividend equals the divisor times the quotient plus the remainder. Now if something is a factor, then that remainder would have to be 0. If it's a factor, it divides in evenly. So if the remainder theorem says if the polynomial is divided by X -, C, then F of C is the remainder. The factor theorem says if that F of C is 0, then X -, C is a factor or it divided evenly. Also, if X -, C is a factor of F of X, then F of C = 0. To do long division with polynomials, we're going to put the polynomial that's on top inside in decreasing order. If there is no power, we should put a zero there as a placeholder. SO4X to the fourth plus zero, X ^3 -4 X squared plus six X + 3. What we're dividing by goes on the outside. So there's the X -, 4 we want to think about X times. What gives me 4X to the fourth? Well, X * 4 X cubed. It's going to be easiest if we keep all the cubes lined up. Now we could think of writing our 4X cubed over here times X -, 4. That helps some students because that shows us that we're really doing distributive property. And that's just 4X to the 4 -, 16 X cubed and a traditional division problem. What we do next is we subtract. So if I have 4X to the 4th -4 X to the 4th, that's going to give me zero. If I have zero X ^3 - -16 X cubed, that's going to give me 16X cubed. Sometimes people are taught to change all the signs once we distribute and then just do a straight addition that works. Also, we're going to then bring down the next term, which is that -4 X squared. And now we're going to ask this X again times what would give me 16X cubed? And the answer is going to be 16 X squared. So if I have sixteen X ^2 * X -, 4, that's going to give me sixteen X ^3 -, 64 X squared. If we wanted to, we could change the signs right now. And then we're just going to add, if we add the positive 16X cubed and the negative are going to cancel -4 X squared and +64 X squared is going to give us 60 X squared, we're going to bring down the next term, So plus 6X. And we're going to ask this X here times, what is going to give me 60 X squared? And the answer's going to be 60 X. So sixty X * X - 4 sixty X ^2 -, 240 X. If we change our signs so we can add the 60X squareds are going to cancel and we're going to get 246 X bring down the three. So the X times, what gives me 246 X 246. So 246 * X -, 4 two 146 X -4 eight 984. So then if we change our signs so we can add, we're going to get 987 as our remainder and our remainder always gets added as the remainder divided by what? We were just taking everything into X -, 4. Now a way that could be checked, we could check this is we could actually multiply the X -, 4 times that whole thing and show that we get out the original. If we multiplied this whole thing out, we would really get out this 4X to the 4th -4 X squared plus six X + 3. I'll let you guys do that. It's just distributive property, so maybe we'll do it together X times. All of it's going to give me 4X to the fourth plus 16X cubed plus sixty X ^2 + 246 X. And then if I take this X -, 4 times the 987 / X - 4, the top and the bottom will cancel. So that would leave me just the 987. Now if I take the -4, I'd get -16 X cubed -64 X squared -240 X -984. If we add those together, we would get 4X to the 4th. The X cubes cancel minus four X ^2 + 1 X -3. Somewhere we had an error here. If we had X, oh, this was 246, I can't read my own writing. There's a 246 then negative or +246 and -240 would have given us 6X, and that does indeed check. That was a plus. Once again, my writing SO4X to the fourth minus four X ^2 + 6 X +3. Now, there's a shortcut method to do this called synthetic division. If we look at the exact same problem and we do synthetic division, we're going to just deal with the coefficients. This time. If there isn't anything, we need to put a zero. So there's a four. There's a zero for the cube, a -4 for the square, A6 for the linear, and a three for a constant. We're going to think about what makes this denominator go to zero and we're going to put it in a little half box off to the left and it's going to be a four. And the steps are going to be always bring down the very first number and write it below the line. So we're going to put a four down there. If we're below the line, we're going to multiply and put it in the next spot over above the line. So 4 * 4 would give me 16. If we're above the line, we're going to add. If we're below the line, we're going to multiply. So four times the 16, the number in the box times the number below the line, gets put in the next column over above the line. If we're above the line, we're going to add so -4 and 64 is 60. If we're below the line, we're going to multiply by the number in the box, SO240. If we're above the line, we're going to add 246. If we're below the line, we're going to multiply 48, 984. If we're above the line, we're going to add 987. Now the way we read this answer, we can do it a couple different ways. If the highest degree was an X to the fourth, our answer, if we're dividing by a linear has got to be the next power lower. So it's going to be four X ^3 plus sixteen X ^2 + 60 X plus 246 + 987 / X -, 4. If we look, that should be exactly what we got before. Four X ^3 + 16 X squared plus sixty X + 246 + 987 / X -, 4. So synthetic division is a shortcut for long division. It works if the denominator has a leading coefficient of one and is a linear term, a polynomial like F of X = 6, X to the fifth minus two X ^3 + 4, X squared minus three X + 1. We want to find F of two using synthetic division. Now remember that F of X is the same thing. So we know that our X is the number 2. So 2 is going to go in the box. Like our long division. We had X -, 2. We could think about taking that and making it equal to 0. So X equal 2. We're going to put all of our coefficients in descending order, remembering to put a zero. If we don't have a power for our X the very first time, we're always going to bring the number to below the line. If we're below the line, we're going to multiply and put the result in the next column over above the line. If we're above the line, we're going to add. If we're below the line, we're going to multiply. Above the line, we're going to add. Below the line, we're going to multiply. Above the line, we're going to add. Below the line, we're going to multiply. Above the line, we're going to add. Below the line, we're going to multiply. Above the line we're going to add. So F of two is 187. We know another way to do this. We could actually stick into 6 * 2 to the 5th -2 * 2 ^3 + 4 * 2 ^2 - 3 * 2 + 1. Well, 2 to the 5th is 32, two cubed is 8/2 squared is 4. Then we've got a bunch of multiplication to do, a 192 -, 16 + 16 - 6 + 1, so F of 2 is going to equal 187. Two different ways to do the same thing. Synthetic division is frequently easier. Thank you and have a wonderful day.