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Introduction to factoring higher degree polynomials

Unpack the process of factoring monomials in algebra. Learn to simplify third-degree polynomials and tackle fourth-degree monomials. Understand the structure of introductory algebra and apply it to higher degree polynomials. Explore the concept of factoring multiple times and delve into the difference of squares. It's all about breaking down complex expressions into simpler parts!

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Video transcript

- [Narrator] When we first learned algebra together, we started factoring polynomials, especially quadratics. We recognized that an expression like x squared could be written as x times x. We also recognized that a polynomial like three x squared plus four x, that in this situation both terms have the common factor of x and you could factor that out and so you could rewrite this as x times three x plus four. And we also learned to do fancier things. We learned to factor things like x squared plus seven x plus 12. We were able to say, "Hey! What two numbers would add up to seven, and if I were to multiply them I'd get 12, and in those early videos, we show why that worked, and we'd say well, three and four, so maybe this can be factored as x plus three times x plus four. If this is unfamiliar to you, I encourage you to go review that in some of the introductory factoring quadratics on Khan Academy. It should be review at this point in your journey. We also looked at things like differences of squares. X squared minus nine. We'd say "Hey, that's x squared minus three squared, so we could factor that as x plus three times x minus three. And we looked at other types of quadratics. Now, as we go deeper into our algebra journeys, we're going to build on this to factor higher degree polynomials. Third degree, fourth degree, fifth degree, which will be very useful in your mathematical careers. But we're going to start doing it by really looking at some of the structures, some of the patterns that we seen in introductory algebra. For example, let's say someone walks up to you on the street and says, "Can you factor x to the third plus seven x squared plus 12 x?" Well, at first your wanna say "Oh, this is a third-degree polynomial that seems kind of intimidating", until you realize, "Hey, all of these terms have the common factor x, so if I factor that out, then it becomes x times x squared plus seven x plus 12." And then, this is exactly what we saw over here, so we could rewrite all of this as x times x plus three times x plus four. So we're going to see that we might be able to do some simple factoring like this, and even factoring multiple times. We might also start to appreciate structure that brings us back to some of what we saw in our introductory algebra. So, for example, you might see something like this, where, once again, someone walks up to you on the street and says "Hey, you factor this, a to the fourth power plus seven a squared plus 12," and at first you're like "Wow! There's a fourth power here, what do I do?" Until you say "Well, what if I were to rewrite this as a squared squared plus seven a squared plus 12." And now, this a squared is looking an awful lot like this x over here. If this were an x, than this would be x squared. If this were an x, than this would just be an x. And then these expressions would be the same. So when I factor it, everywhere I see an x, I could replace with an a squared. So I could factor this out really looking at the same structure we have here as a squared plus three times a squared plus four. Now, I'm going really faster, this is really the introductory video, the overview video. Don't worry if this is a little bit much too fast. This is really just to give you a sense of things. Later in this unit, we're going to dig deeper into each of these cases. But just to give you a sense of where we're going, I'll give you another example that builds off of what you likely saw in your introductory algebra learning. So, building off of the structure here, if someone were to walk up to you again, a lot of people are walking up to you, and say "Factor four x to the sixth minus nine y to the fourth." Well, at first, this looks quite intimidating, until you realize that "Hey, I could write both of these as squares. I could write this first one as two x to the third squared minus, and I could write this second term as three y squared squared." And now, this is just a difference of squares. So it'd be, two x to the third, plus three y squared times two x to the third minus three y squared. We'll also see things like this where we're going to be factoring multiple times. So, once again, someone walks up to you in the street and they, you're a very popular person. Someone walks up to you on the street and says "Factor x to the fourth minus y to the fourth." Well, based on what we just saw, you could realize that this is the same thing as x squared squared minus y squared squared, and you say "Okay, this is a difference of squares, just like this was a difference of squares." So it's going to be the sum of x squared and y squared, x squared plus y squared, times the difference of them. X squared minus y squared. Now this is fun, because this too a difference of squares. So we can rewrite this whole thing as, I'll rewrite this first part, x squared, x squared plus y squared, and then we can factor this as a difference of squares, just as we factored this up here. And we get x plus y times x minus y. So I'll leave you there. I've just bombarded you with a bunch of information, but this is really just to get you warmed up. Don't stress about it because we're gonna go deep into each of these and there's gonna be plenty of chances to practice it on Khan Academy to make sure you understand where all of this is coming from. Enjoy.