 Welcome back to our lecture series Math 1050 College Algebra for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. In lecture 26, we're going to begin a long journey about factoring polynomials. And we've done this for quadratic polynomials already. Factoring them helped us solve quadratic equations. And so we want to do this for higher degree polynomials as well. And so I should mention that the basic factoring techniques we've learned from previous encounters apply. So whenever you're trying to factor a polynomial, I guess the first thing that you mentioned, if you have a polynomial equation, just like quadratic equations, you're going to want to set one side, typically the right-hand side equal to zero. That's always the first step here. Set will say the right-hand side equal to zero. And so for us that means we'd subtract the four x squared from both sides. So our equation x to the fourth equals four x squared becomes x to the fourth minus four x squared equals zero. So once the right-hand side is equal to zero, if we can factor the left-hand side, then we can apply the zero product property to help us solve this equation. And so that's where we're going to begin to factor at this moment. The first thing to do whenever you're factoring, I would say the first thing to do is you're going to look for the GCD, the greatest common divisor. If you look at the coefficients, you have one versus a negative four. One's the only thing common there. But when you look at the powers of x, you do have a x to the fourth and you have an x squared. When you're looking for the GCD of these exponents here, you're looking for the lowest power. You can't take more away than what the least member has right there. And so we can factor out an x squared in this situation. That's what our GCD is going to turn out to be. So factoring out the x squared, you would leave behind x squared minus four equals zero. So we've taken out the GCD. We then want to factor x squared minus four. Then we have to remember the previous factorization formulas we had before. In particular, x squared minus four, that's a difference of squares. We can factor that one as x minus two and x plus two equals zero. Then by the zero product property, we can set each and every one of these factors equal to zero. Setting the first factor equal to zero x squared would give us x equals zero. Setting the second factor equal to zero x minus two, we'd get x is equal to two. And setting the third factor equal to zero, we'd get x plus two equals zero. That is x is negative two. And so we have three roots to this polynomial. Notice that x squared is a repeated root. It's multiplicity would be two. The multiplicity of the other ones is one. So there is a consequence there. But the solutions to this equation are zero, two and negative two. I do want to mention that a temptation many people have in this situation is x to the fourth equals four x squared. If you have this equation, the temptation many people have is actually not to factor out the GCD, but to divide by it. If you divide by x squared, you then end up with simplifying this thing. You'd be getting x squared equals four, taking the square root, you get x equals plus or minus two, which agrees partially with our solution right here. We got the plus or minus two, but where did the zero go? The problem is when you divide by something like x squared, you're making an assumption. When you divide by x squared, you're saying x squared doesn't equal zero, which is actually one of the possibilities right here. If x squared equals zero, that tells us that x equals zero, giving us one of the roots here. So when you divide, you're basically making the assumption that x squared doesn't equal zero, and you should be cautious about that. Whenever you divide, you're making an assumption that something's not equal to zero, which it very well could be, and it's part of the solution. So when it comes to working with polynomials, the solution process is to factor it, not necessarily divide. Now the two processes are very much the same. I like to think of it as the difference between factoring and division as buying ice cream. When you go to the grocery store, this ice cream is typically sold in one of two ways. It's sold in a paper carton, or it's sold in like a plastic bucket. My local grocery store refers to these buckets as a party pale, which I think puts a nice image in our head, right? I love to eat ice cream all the time. And so the difference here is that when you buy your ice cream in a paper carton, once you're done with the ice cream, it was delicious, then you have this sticky paper carton that has no benefit. You can't really clean it out because it's paper, it would just fall apart. So you have no choice but to throw it away. So with the paper ice cream, you eat it and then throw it away. On the other hand, when you buy the party pale, it's made of plastic. Once you're done eating the ice cream, the ice cream parts taste the same, but when you're done, you have this plastic bucket, which if you clean it out, you actually could use it to store lots of things. You could store beans in it, you could store rice, you could store marbles, whatever you want to put in a bucket. I don't care, I'm not judging. I put rocks in them many times in my backyard as I'm trying to clean out my yard and things like that. The thing is once you've eaten the ice cream, there is still utility left with the bucket itself. So what does this have to do with division versus factoring? So in either case, whether you're dividing or you're factoring, you'd be recognized that x squared was common to both terms. The x to the fourth and the four x squared both have an x squared involved there. And so we recognize we have to divide it out. The problem is with division, once you divide out the x squared, you throw it away. You're assuming x squared is not equal to zero. You have a paper card and you just throw it away when you're done. On the other hand, when we factored out the x squared, we don't throw it away when we're done. We recognize oh, x to the fourth and four x squared have that common divisor of x squared, but we don't throw away the card when we're done with it. We take our party pale and we use it to help us solve the problem. So factoring and division are basically the same thing. It's just when you divide, you throw the thing away at the end. With factoring, you keep it recognizing that it leads to part of the solution. So try to factor instead of dividing, but admittedly, the two processes have a very similar calculation. Let's look at another example of a factoring technique that does show up. This one doesn't show up typically for quadratic polynomials, but it's a big deal for higher degree polynomials. Let's take this x cubed minus x squared minus four x plus four. It's already set equal to zero, so we don't have to move anything to the right-hand side. So the next thing to do is once you've done your GCD search, which we look, x cubed negative x squared negative four x plus four, the only thing that's common to all four terms is positive one. The next thing to do really is to look for some special factorization formula, special factoring techniques. So we've seen things like this before, the difference of squares, which we used in the previous example, a perfect square trinomial or the perfect square trinomials we've talked about before. Another factorization formula that we could use, it doesn't apply in this situation nonetheless though, is that if you have a difference of cubes, x minus a cubed, this would multiply out to be, I'm sorry, I'm doing the wrong one. What I wanted to do was the following, sorry, I don't need to foil that thing out. I want to take x cubed minus a cubed. If this thing will factor as x minus a times x squared, whoops, sorry about that, x squared plus a plus a squared, like so. And so this is the difference of cubes factorization. It gets its name like the difference of squares because you have a perfect cube and a perfect cube right here, and a difference of cubes because, well, you're subtracting these things, x cubed minus a cubed. Now the factorization always looks like the following. It's kind of nice here. And I did forget it should be x squared plus a x plus a squared. Sorry about that again. So we have x minus a as the first factor. x minus a basically looks like the thing you started off with, but you forgot the cubes. So we forgot the cubes. That's the first factor. Then the second factor is going to be an irreducible quadratic. Looks like x squared plus a x plus a squared. Now this factor right here is going to look like a perfect square trinomial with the one exception is that the difference between this and that is that the two is missing. So it's kind of funny because when it comes to factorization, many students don't remember the difference of cubes factorization. Honestly, I'd say put it on a no card or something. So you always have it on hand. But to remember the difference of cubes factorization, you're going to forget the cubes, right? So we forget the cubes and you forget the two. So the way to remember is to forget. Forget the cubes, forget the two, and you get the difference of cubes right there. Also, there's a sum of cubes factorization, x cubed plus a cubed. This factor says kind of the same thing. You get x plus a, whoops, I forgot the cubes, and then you get x squared minus a x plus a squared. You forgot, you forgot the two again. Now the important thing here is that the signs on the linear factor and the quadratic factor need to be opposite. You have a plus minus versus a minus plus, that needs to be the case right there. And the sign where you forgot the cubes will match up with that one right there. So you have the sum and difference of cubes formula that can be helpful in factoring cubic polynomials. Unfortunately, that doesn't apply in this situation because difference of cubes would only apply for binomials. We have four terms there. And so the next factoring technique I'm going to suggest here is actually our factoring by groups, which we talked about this with our reverse factoring, our reverse foil technique before. So what we're going to do is we're going to put these things into groups. So we have x cubed minus x squared in the first group, and we're going to have negative 4x plus four in the second group. Take out the GCD of the groups. The first one, you have a GCD of x squared, which we take out, leaving behind x minus one. With the second group, we're going to take out a negative four. Whenever this leading term is negative, always take out the negative sign. So we can see a common factor of four, but we also have a negative sign. So take out a negative four, that leads behind x minus one, the sign changes to equals zero. Then we check the thing that was left behind is actually equal. So we can factor out this common divisor of x minus one, giving us x squared minus four times x minus one. The x squared minus four is a difference of square, so it factors a little bit more. We get x minus two, x plus two, and x minus one. And thus, setting each of these equal to zero, our solutions would be two, negative two, and one. And so this illustrates some of the elementary factoring techniques. What we want to do in this lecture is also develop what's step three, right? We take out the GCDs, we look for special factoring techniques, like the factoring by groups or difference of cubes or things like that. When those don't work, then we resort to a more advanced factoring technique, which we will start developing in the next video. Take a look at it then.