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Rational Roots Theorem - 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 the rational root theorem to find the potential zeros of a polynomial. So for this particular example, we are looking at a fourth degree polynomial and I just want to find what could be my potential rational zeros.

So looking at that, we need to figure out the factors of the last term or the factors of the first term. The terms in the middle don’t really make a difference if we’re finding at least the potential zeros. So it can be plus or minus, factors of the last term, 2, only factors are 1 and 2, over factors of the first term 6, so that’s factors of 1, 2, 3, and 6.

And then we have any sort of combination of these numbers in this fraction form. To some teachers it will be perfectly acceptable just saying these little weird fraction, other teachers are going to want you to write them all out. So let’s just write them all out just to have that extra practice.

So when I do this, I just sort of pick a number on the denominator and then do each pairing of the numerator and go down the row. So we can have 1 over 1, and for this I’m just going to make our life a little bit easier and just say, plus or minus 1, knowing that +1 or -1 could work for this particular instance.

We also have 2 over 1, so plus or minus 2 is an option as well. Okay, going down the row, we have 1 over 2, so plus or minus 1/2, or 2 over 2. But 2 over 2 is 1, which we already have right here, so there’s no need to write it again. So that we did, we have 1 and we have 2, continuing down the list, plus or minus 1 over 3 and plus or minus, plus or minus 2 over 3. And lastly, pairing it with a 6 we have 1/6 or 2/6, but 2 over 6 is the same as 1 over 3 which we already have listed right here.

So using our rational root theorem, factors of the last term over factors of the first theorem, we were able to figure out all potential rational zeros for this polynomial.

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