Introduction to factoring higher degree polynomials | Algebra 2 | Khan Academy

When we first learned algebra together, we started factoring polynomials, especially quadratics. We recognized that an expression like ( x^2 ) could be written as ( x \times x ). We also recognized that a polynomial like ( 3x^2 + 4x ) had the common factor of ( x ), and you could factor that out. So you could rewrite this as ( x \times (3x + 4) ).

We also learned to do fancier things. We learned to factor expressions like ( x^2 + 7x + 12 ). We were able to say, "Hey, what two numbers would add up to ( 7 ), and if I were to multiply them, I get ( 12 )?" In those early videos, we showed why that works and concluded, "Well, ( 3 ) and ( 4 )." So maybe this can be factored as ( (x + 3)(x + 4) ). 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, such as ( x^2 - 9 ). We said, "Hey, that's ( x^2 - 3^2 )," so we could factor that as ( (x + 3)(x - 3) ). 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 by really looking at some of the structure and some of the patterns that we've seen in introductory algebra.

For example, let's say someone walks up to you on the street and says, "Can you factor ( x^3 + 7x^2 + 12x )?" Well, at first, you might 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^2 + 7x + 12) ).

And then this is exactly what we saw over here, so we could rewrite all of this as ( x \times (x + 3)(x + 4) ). So we're going to see that we might be able to do some simple factoring like this and even factor multiple times. We might also start to appreciate structures that bring us back to some of what we saw in our introductory algebra.

For example, you might see something like this, where once again someone walks up to you on the street and says, "Hey, can you factor ( a^4 + 7a^2 + 12 )?" At first, you're like, "Wow, there's a fourth power here; what do I do?" Until you say, "Well, what if I rewrite this as ( (a^2)^2 + 7a^2 + 12 )?"

Now, this ( a^2 ) is looking an awful lot like this ( x ) over here. If this were an ( x ), then this would be ( x^2 ); if this were an ( x ), then this would just be an ( x ); and then these expressions would be the same. So when I factor, everywhere I see an ( x ), I could replace it with ( a^2 ). So I could factor this out, really looking at the same structure we have here as ( (a^2 + 3)(a^2 + 4) ).

Now I'm going really fast for this. This is really the introductory video, the overview video. Don't worry if this is a little 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—lots of people are walking up to you—and say, "Factor ( 4x^6 - 9y^4 )," well, 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 ( (2x^3)^2 - (3y^2)^2 ).

And now this is just a difference of squares. This would be ( (2x^3 + 3y^2)(2x^3 - 3y^2) ). We'll also see things like this where we're going to be factoring multiple times. So once again, someone walks up to you on the street and says, "Factor ( x^4 - y^4 )." Well, based on what we just saw, you could realize that this is the same thing as ( (x^2)^2 - (y^2)^2 ).

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 and difference: ( (x^2 + y^2)(x^2 - y^2) ). Now, this is fun because this is a difference of squares, too. So we can rewrite this whole thing as—I'll rewrite this first part—( x^2(x^2 + y^2) ), and then we could factor this as a difference of squares just as we factored this up here, and we get ( (x + y)(x - 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 going to go deep into each of these, and there's going to be plenty of chances to practice it on Khan Academy to make sure you understand where all of this is coming from. Enjoy!