Hello, wonderful mathematics people. This is Anna Cox from Kellogg Community College. Synthetic division. For our examples of this, we're only going to have A1 coefficient in front of the polynomial that we're dividing by, and it's going to be a shortcut method for long division. So the first thing we're going to do is think about what number makes the second polynomial go to zero. And we're going to put it in a little half box to identify that it's a little separate. And then we're going to write the coefficients in descending order for the polynomial that came first. So one -4 negative two and five. The steps are going to be to always, always bring down the first number. Then if we're below the line, we're going to multiply by the number in the box and we're going to put the result the next line over For above the line we're going to add. We're below the line again, so we're going to multiply by the number in the box and put it in the next column over for above the line, we're going to add below the line, we're going to multiply above the line we're going to add. Now if this is an X ^3 and we divide by an X, we know that our next or our result would start with an X ^2. So the way we read our solution is this is One X ^2 -, 3 X -5 plus whatever that zero is over what we were dividing by. This would be our remainder every time. Truthfully, in this example, we don't need the 0 / X -, 1, but I want you to know what the remainder would look like. Let's look at another example. So in our little half box is going to be a four because it's what makes the second polynomial go to 0. Descending order three -10 negative 9/15. We bring down the first number for below the line we multiply and we put it above the line. The next column over above the line we add. Below the line, we multiply. Above the line, we add. Below the line, we multiply. Above the line we add. If it was an X ^3, 1 power below is going to be an X ^2. So our answer is going to be three X ^2 + 2 X -1 + 11 / X - 4. This next one -2 goes in our box 2 negative 3. There are no X's, so we're going to put a 0 as a placeholder. The first number always goes down. Multiply, add, multiply, add, multiply, add. So we get two X ^2 -, 7 X plus 14 -20 / X + 2. We can do it with fractions. So in our box we're going to have -1 third three descending order. So pay attention because it doesn't always give it to us in descending order. So X ^3, X ^2 X in constant, bring down the first number three times. Negative 1/3 is -1, add, multiply, add, multiply and add. So we get three X ^2 + 6 X -3 + 2 / X + 1/3. Now, frequently we don't like leaving complex fractions, so we could think about multiplying this by 3 / 3 if we so desired to get three X ^2 + 6 X -3 plus 6 / 3 X plus one. Now it's no longer a complex fraction. Our last example before we give you a few to try is to actually solve using synthetic division. So in this case, we're actually going to find F of -3 and -3 is going to go in our half box, because we could think of this as F of X. So X is already the number -3, whereas in the previous examples when we had the X + 1/3, we were really thinking of setting it equal to 0 and solving so -3 goes in our box. Descending order five there is 12. There is no X ^2 X constant. Bring down the first number, multiply, add, multiply, add, multiply, add, multiply, add F of -3 is the number six. Now we could do this a different method. We could say F of -3 and stick -3 in every time we see our unknown and what happens is the numbers get really big really quick -3 to the 4th is 81 five times 81 is 400 and 512 * -27 I don't know off the top of my head. 4 carry two negative 84 plus 9027124 carry 15722324 if I did it right quickly. Wow, these are big numbers and they're way harder than doing the synthetic division. Now. We should still get the same answer. Four O 5 - 324 is going to give me 79. Nope, that's not right. One Wow 81 -, 84 + 981 -. 84 is -3 + 9 is 6. I know which method I would choose. I liked up there way better. I could do it easily in my head. So we have some examples for you to try next. Thank you and have a wonderful day.