Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Synthetic division is a shortcut method of doing long division. When our second polynomial has a leading coefficient of one, we can put the second term thinking about equaling it to 0. So X equal 1, we're going to put the number 1 1/2 box. Then we're going to write the coefficients one -4 negative two and five. Our first step in synthetic division is always to bring down the first number. So we're going to bring down the one. We're going to multiply by the number in the box. So 1 * 1 is 1 and put it in the next column over. Then we're going to add if we're above the line. If we're below the line, we're going to multiply by the number in the box. So -3 * 1 is -3. If we're above the line, we're going to add. If we're below the line, we're going to multiply. Put it in the next column above the line. If we're above the line, we're going to add. Now the way we read this answer is this first number one underneath the line is going to have an X to 1 less power than the original polynomial. So our answer is going to be One X ^2 -3 X -5. That zero would have been our remainder. If we had a remainder, we would have put plus 0 / X -, 1. The fact that it's a 0 remainder tells us it divides evenly into it or it's a factor. This next one, we're going to put four in our half box because that X -, 4 would equal 0X would equal 4. We're going to write three -10 negative 9 and 15. We're always going to bring down the first number. Multiply by the number in the box, add, multiply by the number in the box, add. Multiply by the number in the box, add. So our answer to this one's going to be three X ^2 + 2 X -1 + 11 over X -, 4. If we have some terms that are missing, we're going to do it the same way, but we're going to need to put a placeholder of a zero. So we're going to put a -2 in the box and we're going to get two negative 30 and eight. Bring down the first number. When we're below the line, we multiply and put it above the line. Add, multiply, add, multiply, add. So our answer here is going to be two X ^2 -, 7 X plus 14 + -20 / X + 2. This next one, we're going to put 1/3 in our box. We're going to make sure we put it in decreasing order. So X cubes, then X squareds, then X's, then constants. Bring down the 1st #3 * 1/3 is going to be 10. It was negative 1/3 because if we have that X + 1/3 equaling 0X is going to equal -1 third. So bring down the number 3 * -1 third is -1. When we add we get six 6 * -1 third is -2. When we add, we get -3 -3 * -1 third is 1. When we add we get 2. So our answer to this is going to be three X ^2 + 6 X -3 + 2 / X + 1/3. Now we can use synthetic division to find FA -3. We already know one method, and that's to put the -3 every time we see our X. Sometimes that gives us really large numbers, though, and it's good to know alternative methods. So we're going to put the -3 directly in the box, because if this had been F of X, the X would be the -3. We're going to put 5 and 12 zero because there was no X squareds. 28 and 9 do our synthetic division just like we've been doing. So we're going to multiply and add, multiply and add. When we get done, our F of -3 is just going to be whatever this remainder is. So F of -3 is 6. So this is really, really, really a point. It's the point when X is -3 Y is 6. Because if we did our division, the remainder would be how far above or below the X axis we would be if we were thinking about graphing it. This next one we would put the four in the box one -611 negative 1720. We bring down the first number. We multiply, we add, we multiply. We add, we multiply, we add, we multiply. In this case, it's actually got a remainder of 0, so our F of 4 = 0. This is really, really a point. It's the .40. The fact that the remainder is 0 would tell us that it would be located on the X axis if we were graphing this. Thank you and have a wonderful day.