❖ Synthetic Division – A Shortcut for Long Division! ❖

In this video we're gonna do an example of what's called synthetic division Synthetic division is when you're basically dividing a polynomial by a first degree polynomial so x to the first power it may or may not have a number in there if it was just x you would just break things up individually that'd be easy anyway So suppose we wanna divide, again, x cubed minus 2x squared plus 3x minus 4 we're gonna divide that by (x-2) What you do in this case is this Whatever we're dividing by…

So notice there's a minus 2 — it's x minus 2 we actually take the opposite sign and use positive 2 and that's what you stick outside And then notice I have 1 x cubed minus 2x squared plus 3x minus 4 Notice there's no terms missing. It starts from the third degree second degree first degree and then we just have a number All we do is write the coefficients So there's a 1 a negative 2 in front of the x squared so a positive 3 in front of the x and there's a negative 4 just hanging out And with synthetic division what you do is the first term you just drop it right down there's nothing to do And then the idea is we multiply So I'm gonna take positive 2 times 1, and then stick that number in the next spot So it says postive 2 times 1 will give us positive 2 And then what we do is we add those numbers together in this column So negative 2 plus 2 is gonna give me zero And then we repeat this process Now we take positive 2, multiply by zero and then we stick it in the next column So it says positive 2 times zero is just gonna give us zero So if we add those together we get positive 3 And then again I take positive 2 times positive 3 I'm just gonna erase my little red dots here Take positive 2 times positive 3, that'll give me positive 6 We add the column together and we get the number 2 This last number is gonna be your remainder If I could spell correctly here And what it says is, in summary, it says We can actually — So the numbers in the bottom are what are important 'Kay so it says we can actually rewrite x cubed minus 2x squared plus 3x minus 4 divided by (x-2) So I'll just keep my numbers at the bottom It says we can actually write that as — Okay so the highest power is x cubed and to make things one degree less so this is gonna go with my x squared term this is gonna go with my x term this is gonna be my constant and again the other part is my remainder It says we can actually write x cubed minus 2x squared plus 3x minus 4 divided by (x-2) as 1 x squared plus zero x So usually zero x we'll leave out so I'll erase it So 1 x squared plus 3 and then my remainder is 2 and we put that over what we were originally dividing by which is (x-2) And now we have broken up our original fraction into something a little bit more simple So this would be your solution after you did the synthetic division So a useful little trick One other thing I wanna point out about synthetic division Notice that if we look at the thing on top If I call that f(x) So if I say f(x) equals x cubed minus 2x squared plus 3x minus 4 Suppose I wanted to evaluate this — Suppose I want to plug in the number 2 Well it turns out if you plug in the number 2 If you do the synthetic division with (x-2), if you do that synthetic division it says whatever the remainder is that is actually gonna be the solution when you plug that value in So just a coincidence here that I plug 2 in and got 2 out But notice if I plug 2 in I'll get 2 cubed minus 2 times 2 squared plus 3 times 2 minus 4 So this is 8 minus 8 plus 6 minus 4 Well the 8 and the 8 cancel out positive 6 minus 4 is, hey, positive 2 Okay, so synthetic division is actually also a way to evaluate polynomials That remainder, again, is gonna be your solution And sometimes if you have a real tedious problem, using synthetic division can be useful For example if you're plugging things into a calculator Typically I don't use it, really hardly ever but that's just a personal preference of mine so Alright, I hope this example makes some sense I think I'm gonna do one other as well and just kinda reinforce this stuff If you have any questions just send me an email

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