Main content

## Factoring higher degree polynomials

# Factoring higher degree polynomials

CCSS.Math:

## Video transcript

- [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.