Factoring higher degree polynomials | Algebra 2 | Khan Academy

There are many videos on Khan Academy where we talk about factoring polynomials, but 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 warm-up. Let's say that we wanted to factor 6x squared plus 9x times x squared minus 4x plus 4. 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. The way that this might be a little bit different than what you've seen before is that 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, 6x squared plus 9x; both 6x squared and 9x are divisible by 3x. So let's factor out a 3x here. This is the same thing as 3x times 3x times what is 6x squared? Well, 3 times 2 is 6, and x times x is x squared. And then 3x times what is 9x? Well, 3x times 3 is 9x. You can verify that if we were to distribute this 3x, you would get 6x squared plus 9x.

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 4 and whose sum is negative 4,” and you might say, “Hey, that's negative 2 and negative 2.” And so this would be x minus 2. We could write it as x minus 2 squared, or we could write it as x minus 2 times x minus 2.

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 could 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 4x squared plus 6x minus 24. Just like always, pause this video and see if you can have a go at it. 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, so see if you could have a go at that. All right, now let's do it together. You can't always factor a third-degree polynomial by grouping, but sometimes you can, so it's good to look for it.

When we see it written like this, we say, “Okay, x to the third minus 4x 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 6x and negative 24? Yeah, they're both divisible by 6. So let's factor out a 6 here. So plus 6 times x minus 4.

Now you are probably seeing the home stretch, where you have something times x minus 4 and then something else times x minus 4. You can sometimes, I like to say, undistribute the x minus 4 or factor out the x minus 4. So this is going to be x minus 4 times x squared plus 6. And we are done.