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Current time:0:00Total duration:5:53

Introduction to factoring higher degree polynomials

Video transcript

when we first learned algebra together we started factoring polynomials especially quadratics we recognized that an expression like x squared could be written as x times X we also recognized that a polynomial like 3x squared plus 4x than in this situation both terms had the common factor of X and you could factor that out and so you could rewrite this as x times 3x plus 4 and we also learn to do fancier things we learn to factor things like x squared plus 7x plus 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 and in those early videos we show why that works you say well 3 & 4 so maybe this can be factored as X plus 3 times X plus 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 the at this point in your journey we also looked at things like differences of squares x squared minus 9 say hey that's x squared minus 3 squared so we could factor that as X plus 3 times X minus 3 and we looked at other types of quadratics now as we go deeper into our algebra Journeys we're going to build on this 2 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 doing it by really looking at some of the structure 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 to the third plus 7x squared plus 12x well at freshmen 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 squared plus 7x plus 12 and then this is exactly what we saw over here so we could rewrite all of this as x times x plus three times X plus four so we're going to see that we might be able to do some simple factoring like this and even factoring multiple times we might also start to appreciate structure that brings us back to some of what we saw in our introductory algebra so for example you might see something like this where once again someone walks up to you on the street and says hey you factor this a to the fourth power plus 7a squared plus twelve and at first like wow there's a there's a fourth power here what do I do until you say well what if I were to rewrite this as a squared squared plus 7a squared plus twelve and now this a squared is looking an awful lot like this x over here if this were an X then this would be x squared if this were an X then this would just be an X and then these expressions would be the same so when i factor it everywhere I see an X I could replace with an a squared so I could factor this out really looking at the same structure we have here as a squared plus three times a squared plus four now I'm going really faster this this is really the introductory video the overview video don't worry if this is a little bit much 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 a lot of people are walking up to you and say factor 4x to the sixth minus 9y to the fourth what 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 to the third squared - and I could write this the second term as 3y squared squared and now this is just a difference of squares so it'd be 2 X to the third plus three y squared times 2x to the third minus three y squared we'll also see things like this where we're going to be factoring multiple times so once again someone walks up to you in the street and they say you're a very popular person something walks ap'to stays up to you on the street and says factor X to the fourth minus y to the fourth well based on what we just saw you could realize that this is the same thing as x squared squared minus y squared squared 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 of x squared + y squared x squared plus y squared times the difference of them x squared minus y squared now this is fun because this is two a difference of squares so we can rewrite this whole thing as I'll rewrite this first part x squared x squared plus y squared and then we could factor this as a difference of squares just as we factored this up here and we get X plus y times X minus 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 gonna 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