- [Voiceover] So let's say that we've got the polynomial 16x to the third plus 24x squared plus nine x. Now what I'd like you to do is pause the video and see if you could factor this polynomial completely. Now let's work through it together. So the first thing that you might notice is that all of the
terms are divisible by x so we can actually factor out an x. So let's do that, and
actually, if we look at these coefficients, it
looks like, let's see, it looks like they don't have any common factors other than one. So it looks like the
largest monomial that we can factor out is just going to be an x. So let's do that. Let's factor out an x. So then this is gonna be x times. When you factor out an
x from 16x to the third you're gonna be left with 16x squared, and then plus 24x and then plus nine. Now this is starting to look interesting so let me just rewrite it. It's gonna be x times. This part over here looks interesting because when I see the 16x squared, this looks like a perfect square. Let me write it out. 16x squared. That's the same thing as four x squared, and then we have a nine over there, which is clearly a perfect square. That is three squared. Three squared. And when we look at this 24x, we see that it is four times three times two, and so we can write it as,
let me write it this way. So this is going to be plus two times four times three x. So let me make it, so two times four times three times three x. Now why did I take the trouble, why did I take the trouble of
writing everything like this? Because we see that it fits the pattern for a perfect square. What do I mean by that? Well in previous videos, we saw that if you have something of the form Ax plus B, and you were to square it, you're going to get Ax squared plus two ABx plus B squared, and we have
that form right over here. This is the Ax squared.
Let me do the same color. The Ax squared. Ax squared. We have the B squared.
You have the B squared. And then you have the two ABx. Two ABx right over there. So this section, this entire
section, we can rewrite as being we know it, what A and B are. A is four and B is three
so this is going to be Ax, so four x plus four x plus b, which we know to be three. That whole thing, that whole
thing squared, and now we can't forget this x out front
so we have that x out front. And we're done. We have
just factored this. We have just factored this completely. We could write it as x
times four x plus three and then write out and then say times four x plus three or we could just write x times the quantity four
x plus three squared. And so we've just factored it completely, and the key realizations here is well, one: what could I factor
out from all of these terms? I could factor out an x
from all of those terms, and then to realize what we had left over was a perfect square,
really using this pattern that we were able to
see in previous videos.