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Using Synthetic Division to Factor - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

One way that we can factor a polynomial is to use synthetic division. Synthetic division factoring can be an effective way to factor complicated polynomials with whole number coefficients. When factoring using synthetic division, we determine one of the roots and use synthetic division to determine the remaining coefficient. If the remaining coefficient is equal to zero, then the expression is divisible by that factor.

Factoring using synthetic division, okay? So in this we're going to talk about how we can use synthetic division to factor a polynomial. But before I do that, I just want to go into some background about what makes a factor. Okay.
So, first off is 6a factor of 18. Okay? Thinking about it we know that 6 goes into 18, so yes it is. Mathematically what you're actually doing is 18 divided by 6 is equal to 3. There is no remainder so therefore, it is a factor. Okay? If I asked you is 5 a factor of 18. This you know is, it isn't because it's not true because 18 divided by 5 is equal to not a whole number, okay. It's equal to 3 plus a fraction. You already have a remainder left over so it doesn't in fact go in. Okay.
So numbers are easier to deal with, okay? And for a certain extent polynomials aren't that bad as well either, okay? We can say you know. Factor x squared minus 5x plus 6, we know how to factor this. x-2, x-3 so in fact what we have is x-2 as a factor and x-3 is a factor of this polynomial right here. Okay.
Where synthetic comes in, division comes in is if we're dealing with something at a higher degree than 2, we may not know how to factor this out by hand. Okay. If this was an x cubed right here, I wouldn't know where to go and so we would have to do synthetic division to factor it out. Okay?
So main thing we're looking for is in doing synthetic division, we know something is a factor if and only if we get a remainder of zero when we divide, okay? Going back over here, we knew 6 was a factor because we had a when we divided we had 3 no remainder. We know that 5 wasn't a factor because when we divided it, we had a remainder. Okay synthetic division is going to tell us the exact same things. We divide it out we have a remainder, it's not a factor. We don't have a remainder, it is.

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