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Dividing Polynomials using Synthetic Division - Problem 1

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!

Using synthetic division to divide polynomials. So in order to, using that division to divide polynomials, we first have to look at our divisor, our denominator to make sure that if fits the restrictions that we need in order to use this process.

So the first restriction is that our denominator has to be of degree one, we can only have a single x, no x² and anything like that, so this one checks out. The second thing we have to look at is to make sure there is no coefficient on our X term, in this case we don’t, so we can use synthetic division.

How synthetic division works is we look at our denominator and pick the number that would make our denominator equal to zero, in this case -1. Once we have that written down, we make a little bracket and then in the same row that we wrote our -1, we want to write the coefficients for our dividend for our numerator, okay? We want to make sure that each power on x is accommodated for, so start with x³, add coefficient of 1.

Going down the line, we don’t have any x² terms, but we still need a place holder for it, so we need to put a 0 in there, continuing down to our x term, -13 and lastly -12. Okay, now, how synthetic division works, first we want to drop down this 1. Second we multiply the term on the outside by the term we just dropped down so -1 times 1 is -1, and then we add. So you have plus -1, -1. Continue this process all the way down the row, -1 times -1 becomes 1, add -12, -1 times -12 is 12 and our last term in this case is 0.

0 is the remainder so in this case we don’t have anything extra, and then interpreting your results is just going up by powers of X. So this first thing represents our constant term, this is going to be -12, from our constant term we go to our first degree term, this is –x, this turns into our x² which will be x².

We should always make sure because we’re dividing by a single x, your quotient, what you’re left with should always be one degree less than what you started with. So here we started with an x³, it makes sense that our quotient then is a x², okay? So using synthetic division to divide polynomials, going through the process, always make sure you put the number that gives you a 0 outside and then drop down your first term and then add.

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