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# Factoring perfect squares: common factor

## Video transcript

so let's say that we've got the polynomial 16 X to the third plus 24 x squared plus 9x now what I'd like you to do is pause the video and see if you can 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 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 going to be x times when you factor out an X from 16x to the third you're going to be left with 16x squared and then plus 24x and then plus 9 now this is starting to look interesting so let me just rewrite it this is going to 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 4x 4x squared and then we have a 9 over there which is clearly a perfect square that is 3 squared 3 squared and when we look at this 24x we see that it is 4 times 3 times 2 and so we can write it as let me write it this way so this is going to be plus 2 times 4 times 3x so let me make it so 2 times 4 times 3 times 3x now why did I 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 a x squared plus 2a be X 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 a B X 2 a B X right over there so this section this entire section we can rewrite as being we know what a and B are a is four and B is 3 so this is going to be ax so 4 X plus 4 X plus B which we know to be 3 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 4 X plus 3 and then write out and then say times 4 X plus 3 or we could just write x times the quantity 4x plus 3 squared and so we've just factored it completely and the key realization here is will 1 you know what can i factor out from all of these terms I could factor out an X from all of those terms and then to realize that what we had left over was a perfect square really using this pattern that we were able to see in previous videos