College Algebra: The Factor Theorem and Its Uses

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Now, you may remember by the remainder theorem we know that if we want to find out the remainder when a particular polynomial is divided by something of the form x - c, all I have to do to find the remainder is look at the polynomial evaluated at that c. So, for example, if I want to find out the remainder when I have f(x) and I divide it by x - c, the remainder will always be just the polynomial evaluated at c. So that's pretty cool, but in fact, that will allow us to find factors of polynomials, because a factor of a polynomial is just some piece that when you divide it in, the remainder is zero. There's nothing left over. It goes in evenly. That's what it means to be a factor.
So let's take a look at a very simple example and then build the factor theorem. So let's consider the following: x² + x - 6. Now, this is a very, very simple example, but I want to illustrate the method. So with this particular polynomial, what I want to do is I want to see what is the remainder when I divide that by x + 3. Now, I want the remainder. Okay, well, how would I find that? Well, what I could do is just do the long division, or I could use synthetic division and see what the remainder is. I want to remind you of synthetic division now. How do you use synthetic division? You can only use synthetic division when you have a thing that you're dividing down here that looks like x plus or minus a number. That's the only time synthetic division works. A great mistake in life is to try synthetic division with other things--can't do it. What's the rule? You flip the sign of this. So you write a -3. You create sort of half a house. You copy the coefficients making sure everyone's represented. 1, 1, -6, and then you perform the following simple task. You bring down the first number, and then you multiply this number by that, and write the answer here, so this will be -3. And then add -2. And now repeat. Take this product, that's 6, add, and you get zero. And that last term is always the remainder. Hey, look. I see zero. That means that this thing has remainder of zero. That means that, in fact, if I look at this divided by that, the answer is just x - 2. Well, that means this is actually a factor of this, because what I see is x² + x - 6 equals x + 3 times x - 2. You see, it's a factor. This times something gives me that.
So there's a connection between finding a factor and seeing a root--seeing a zero remainder. So if I have a zero remainder, that means this must be a factor. But remember, the remainder is just the value of the function at that point. So, in fact, this must be what we call--well, the zero of the polynomial, which means where this thing equals zero. So, what that means, in summary, if you've got a value for c, which makes this equal to zero--so if you can make this equal to zero, that means that x - c must be a factor. So this must go into here evenly, zero remainder. And the converse is also true. If I've got an x - c that divides in evenly into the polynomial, then the remainder must be zero. So, in fact, this must equal zero.
So, the factor theorem states the following. The factor theorem says, "The polynomial x - k is a factor of a polynomial f(x) if and only if f(k) equals zero." So x - k--this kind of thing--is going to be a factor of a polynomial precisely when this equals zero, and that is the factor theorem. So the bottom line is, if you want to see if something is a factor, all you've got to do is take that number there and plug it in and see if you get zero.
Okay, so to illustrate that, let's play a little teeny game. I don't have time for a long game here. It's time for "Is it a Factor," with your host Ed Burger. So, welcome to "Is it a Factor." Let me remind you how this game is played, especially for those of you who are new to our program, which basically is everybody in the world because I'm just making this up on the fly. The idea is I'm going to give you a polynomial, f(x). I'm going to give you this potential factor, and your job is to use the factor theorem to determine if this genuinely is a factor or not. That's the mission. Let's see how we do. So is this a factor or not? Well, the thing to do is to say, "Okay, that factor theorem tells me this is a factor precisely when the opposite of this number here"--I have to write this in the form x - k, which means I'd have to write this as x - -1, so it's the opposite, it's the -1--precisely when -1 would be a solution to that equals zero. Sometimes we call that the zeros of this polynomial. So all I have to do to see if that's a factor--I don't have to long divide out. All I've got to do is take -1 and plug it in for x and ask, "is that zero?" If it equals zero, this is a factor. If it doesn't equal zero, this is not a factor.
So let's see what happens if I plug in -1 into here. I would see a -2, because -1³ is -1, and then I would see a +1 and a +2. And that, -2 + 1 is -1 + 2 is 1. So, in fact, this equals 1, which is not equal to zero. That's the only question you ask, and therefore not a factor. So this is not a factor. Let's try another one. x³ + 2x² - 3, and I want to know is x - 1 a factor or not. Well, I'm going to use the factor theorem which says all you've got to do is take the opposite sign of that, so take the 1, and plug the 1 in and see if that produces a zero. If a 1 plugged in here produces a zero, it's a factor. If it doesn't produce a zero, it's not a factor. Remember, I plug a 1. I don't plug a -1. Remember, the factor I'm considering is x - k, and to see if it's a factor I plug in just k alone into the polynomial. So you always have to remember to switch that sign. So I look now at 1 and I see 1³, which is 1, plus 2 times 1, which is 2, minus 3. Well, 1 + 2 is 3, -3 is zero. Aha! We have a factor. So that means that x - 1 is a factor of this polynomial. So x - 1 times something else will equal that. So you can see whether we have a factor or not just by, not long dividing now, just by taking the opposite sign, in this case, -1, plugging it in here, seeing if it's zero or not, in this case, taking 1, plugging in here, seeing if it's zero or not, and you can actually find whether this thing is a factor or not by using the factor theorem.
Polynomial and Rational Functions
The Factor Theorem
The Factor Theorem and Its Uses Page [1 of 2]