 In this video I want to offer a shortcut to long division of polynomials that often gets the name of synthetic division. Now synthetic division is a shortcut which is like oh if there was a shortcut why didn't we start with that? Well that's because it's a special case that we can only use synthetic division when our denominator that is the divisor is going to be x minus c so it's going to be x minus a constant. Do you notice the minus there? That's going to be critical and so this is going to be a special case of what we can zip through the division when we divide by x minus c. Now it turns out this special case is going to show up with enough frequency that this shortcut is going to be super super worth it and so the best way to explain synthetic division is actually can just do it with a specific example. We're going to take the polynomial 6x squared minus 26x plus 12 and we're going to divide it by x minus 4. So when you do synthetic division you're going to draw your division box but this time it's going to be upside down and in descending order we're going to record the coefficients of the numerator just the coefficients so we're going to get a 6 we're going to get a negative 26 and we're going to get a 12 like so and so we just write the coefficients and it has to be in descending order so you start with the biggest power then the next power then the next power like so. Notice our divisor is x minus 4 this is exactly the format that we allow x minus c so to the left we're going to write the number 4 because after all we're dividing by x minus c x minus 4 we're just going to record the number c right here so we're going to divide this 4 and then draw draw a horizontal line that cover that's below the 4 and the bracket here but leave a gap because we're going to write some stuff here so now the first step when it comes to the synthetic division you look at the leading coefficient and you're going to drop it down so you're just going to write down a 6 great the next step is we're going to take the number 6 and the number 4 and we're going to multiply them together so we're going to get 4 times 6 which is equal to 24 and then we're going to write that in the next column over which is 24 right here and the next column over then what we do is we're going to take this column and we're going to add things together negative 26 plus 24 this is going to give us a negative 2 we then record that on the bottom then this process repeats itself we're going to take negative 2 and we're going to times it by 4 4 times negative 2 is equal to a negative 8 we then record that in the next column negative 8 you then are going to add the numbers in this column together so we're going to take 12 plus negative 8 that gives us a positive 4 and then I'm going to draw a little box around the 4 to indicate that we are now done and so I want you to so this is this is the tableau we filled out for synthetic divisions like okay what do these numbers mean so when you look at the bottom row before the box these right here are going to be the coefficients of the quotient here the quotient that is the quotient is going to equal 6x minus 2 let me write that in white to make it more official here 6x minus 2 so we take the coefficient 6 and negative 2 but then we're going to downgrade the power by 1 right we started off with a degree 2 polynomial if you divide it by a degree 1 polynomial you'll get a degree 1 polynomial polynomial so the quotient is going to be linear and the coefficients are going to be 6 and negative 2 you get that then the remainder is going to be this number over here the remainder it's going to be 4 and so we're going to get the remainder here is a 4 and so 6x squared minus 26x plus 12 divided by x minus 4 is equal to 6x minus 2 remainder remainder 4 or in other words if we take 6x squared minus 26 26x plus 12 and we divide this by x minus 4 we end up with 6x minus 2 with a remainder of 4 over x minus 4 and so this is how we perform synthetic division now I want you to compare what we did in this example to a previous video for it's the link you can now see on the screen we did this exact same calculation using long division and our quotient was 6x minus 2 and our remainder is 4 is the exact same but synthetic division has a great simplicity to it we can fill out this this tableau super super fast and I want to show you some examples of this so let's take the let's take the polynomial 2x squared plus 5x plus 15 and divided by x minus so we're going to record just the coefficients of the dividend and descending order so we get 2 5 and 15 and then we're going to divide this by we just look at the number we're subtracting x minus 3 so we record a 3 over here so remember the instructions so going right to left or excuse me going left to right in our tableau we drop down the first number we get a 2 2 times our divisor here 3 2 times 3 is a 6 and then we take 5 plus 6 which is 11 then we take 3 times 11 which is 33 and then we take 12 plus 33 which is going to be 45 and this will then be our remainder when we go through this oh it occurred to me that I actually had a 15 right here so I should correct this so that doesn't confuse anyone watching as we had a 15 which then changes the remainder 15 plus 33 was a 48 and so this tells us that 2x squared plus 5x plus 15 divided by x minus 3 is equal to 2x plus 11 where did the 2x plus 11 come from it came from these digits right here we started off with a quadratic polynomial so the quotient will be one less because we divided by a linear polynomial so we get 2x plus 11 and then we add to it the remainder of 48 over x minus 3 and so that's what we get when we do synthetic division it's super super slick super super fast with enough practice you're going to be able to zoom through these things really quickly let's do another example this time let's do a degree 3 polynomial this thing does not really get much more difficult even if the polynomial gets bigger and bigger bigger so we have to write the coefficients of the numerator so we get 1x cubed minus 4x squared 0x minus 5 it's very critical that you put a 0 in for the x term right so there's no x term here it's because you have a 0x you need to put that 0 in there if you didn't do the 0 what this would look like to you is actually x squared minus 4x minus 5 if you don't have the 0 in there by mistake you're actually divided by a quadratic polynomial not a cubic and we're going to divide this by 3 so with synthetic division we get the following bring down the 1 1 times 3 is 3 negative 4 plus 3 is negative 1 negative 1 times 3 is a negative 3 plus 0 is a negative 3 negative 3 times 3 is negative 9 plus negative 5 is a negative 14 that's the last term so that's going to be the remainder so what we've then discovered is the following x cubed minus 4x squared minus 5 divided by x minus 3 this will equal 1x squared minus x minus 3 that there's our quotient the quotient's leading term will always be one less than the leading term we started with and then our remainder will be negative 14 over x minus 3 and there you have it