Factoring higher degree polynomials
- [Instructor] There are many videos on Khan Academy where we talk about factoring polynomials. What we're going to do in this video is do a few more examples of factoring higher degree polynomials. So let's start with a little bit of a warmup. Let's say that we wanted to factor six x squared plus nine x times x squared minus four x plus four. Pause this video and see if you can factor this into the product of even more expressions. All right, now let's do this together and the way that this might be a little bit different than what you've seen before is this is already partially factored. This polynomial, this higher degree polynomial, is already expressed as the product of two quadratic expressions but as you might be able to tell, we can factor this further. For example, six x squared plus nine x, both six x squared and nine x are divisible by three x. So let's factor out a three x here. So this is the same thing as three x times, three x times what is six x squared? Well, three times two is six and x times x is x squared and then three x times what is nine x? Well, three x times three is nine x and you can verify that if we were to distribute this three x, you would get six x squared plus nine x and then what about this second expression right over here? Can we factor this? Well, you might recognize this as a perfect square. Some of you might have said, hey, I need to come up with two numbers whose product is four and whose sum is negative four and you might say, hey, that's negative two and negative two and so this would be x minus two. We could write x minus two squared or we could write it as x minus two times x minus two. If what I just did is unfamiliar, I encourage you to go back and watch videos on factoring perfect square quadratics and things like that but there you have it. I think we have factored this as far as we can go. So now let's do a slightly trickier higher degree polynomial. So let's say we wanted to factor x to the third minus four x squared plus six x minus 24 and just like always, pause this video and see if you can have a go at it and I'll give you a little bit of a hint. You can factor in this case by grouping and in some ways it's a little bit easier than what we've done in the past. Historically, when we've learned factoring by grouping, we've looked at a quadratic and then we looked at the middle term, the x term of the quadratic and we broke it up so that we had four terms. Here, we already have four terms. See if you can have a go at that. All right, now let's do it together. So you can't always factor a third degree polynomial by grouping but sometimes you can so it's good to look for it. So when we see it written like this, we say okay, x to the third minus four x squared, is there a common factor here? Well, yeah, both x to the third and negative four x squared are divisible by x squared. So what happens if we factor out an x squared? So that's x squared times x minus four and what about these second two terms? Is there a common factor between six x and negative 24? Yeah, they're both divisible by six. So let's factor out a six here. So plus six times x minus four and now you are probably seeing the homestretch where you have something times x minus four and then something else times x minus four and so you can, sometimes I like to say undistribute the x minus four or factor out the x minus four and so this is going to be x minus four times x squared, x squared plus six and we are done.