Okay, so last time we ended with this polynomial and So we're gonna hope so the hope is to get this into some type of a Plus B whole squared form in the final factorization So let's see what we can do You can think of this as X squared plus, well we'll fill the middle term in just a second, 16 is 4 squared. So we can think of this as a and think of this as b and recall that a plus b squared is a squared plus 2ab plus b squared. So we're thinking that hopefully, or just maybe, this might be of the form 2ab, i.e., 2 times x times 4 because b is 4 and indeed you can see 2 times 4 this is 8 so we have 8x.
So indeed this is of the form x squared or a squared plus 2ab plus b squared. So hence this is equal to x plus 4 squared because again this was x squared and this was 2 times x times 4 and this 16 was 4 squared okay all right so this was what we call a perfect square We're going to, but what about if we change things just a bit? What if we had something like x squared plus 10x plus 16?
Well, this is now no longer, this is definitely no longer of the form x plus, I mean it's not equal to, or actually it's not of the form, but it's not, it's no longer equal to, no longer equal to x plus 4 quantity squared, right? Because... x plus 4 quantity squared was x squared plus 8x plus 16. So what do we do with something like this?
Well it turns out in a situation like this we can still factor. So this is now a more, we're moving to the more general factoring method for quadratic polynomials. I'll just call it quadratics. What is that method? Well, what we do is we take the coefficient of x squared, which is 1 in this case, we're going to multiply that by 16, by the constant term.
So 16 times 1, I'm going to put up here, I'm going to make a little diamond, we're going to call this the diamond method. And then there's the middle term, which is 10. I'm going to put that down here. And the hope is, if we're going to be able to factor in a nice way, we're going to need two numbers. Let's call it n and m.
We're going to need two numbers such that n times m is 16, because 16 times 1 is just 16. And n plus m, the sum of those, is 10. So we need two numbers. Essentially these two numbers multiply to 16 and they add to 10. Well, what are some numbers that multiply to 16? Well, 4...
times 4 is 16 but 4 plus 4 is not 10 so 4 and 4 doesn't quite work out but if we choose 8 and 2 8 times 2 is 16 and 8 plus 2 is 10. So n here, and you can choose either to, you know, one of them is going to be 8, one of them is going to be 2, and what we're going to do is we're going to take this middle term 10x, so we have x squared plus 10x plus 16, we're going to rewrite this 10x as 8x plus 2x because we have 8 and 2 that sort of worked out in this diamond. And we'll see what to do with it in just a second. Okay, so we're going to rewrite this as x squared plus 8x plus 2x plus 16. And notice this is still the same polynomial, it's just that we split up the 10x into 8x plus 2x.
And now what we're going to do is we're going to do kind of what we did in the very beginning of the last video, which was a linear factoring. We're going to factor two terms at a time and just take out what's in common. So we're going to factor two terms at a time, simply removing the greatest common factor. Removing the greatest common factor, GCF, greatest common factor.
So for example, in x squared plus 8x, what's in common is just x. So we can take out the x and write it as x times x plus 8. Note that if you multiplied x times x you would get back x squared. If you multiplied x by 8 you would get back 8x. So the factorization of these first two terms is x times x plus 8. Now here we have 2x plus 16. What's in common there?
Well, 2x plus 16 is, what's in common is just a 2. 2 times x. x plus 8. So again if you multiplied 2 to the x you would get the 2x, if you multiply 2 to the 8 you would get 16 back. So overall, all in all, let me rewrite the whole thing. When we go, when we return to the original guy here we can rewrite this as x times x plus 8 plus 2 times x plus 8. because of the fact that x squared plus 8x is x times x plus 8. 2x plus 16 is 2 times x plus 8. And now we're in good shape. We're almost done.
The last step here is to recognize that now this is in common. This x plus 8 is in common. So how would we factor this? Well, we would take out the x plus 8. We can rewrite this then.
as x plus 8 times x plus 2 or x plus 2 times x plus 8, whichever way you wanted to write it. Notice, how do we know that this is true, this actually works out? Well, notice if we, just to check here, we're actually done with the problem now.
The original polynomial x squared plus 10x plus 16 is actually equal to this factorization, x plus 2 times x plus 8. How do we know that this is actually, that these two are equal? Because this tends to be something that kind of trips people up initially. Well, if you multiplied it back, how would you multiply it back? Well, you would take the x and multiply by this, x times x plus 8. And then you would take the 2 and multiply by the other parentheses, plus 2 times x plus 8. And that's exactly this expression above. So indeed, these are equal.
So that's the more general method of factorization. And eventually you may be able to skip some of these in-between steps, especially for these cases where you just have an x squared term and no other coefficient. But in the beginning, it's good to do everything step by step.
All right, let's do one last example and then we'll end this video there. We're going to look at 12x squared. minus 26x minus 10. So in these examples it's always good to first just take out a constant that's in common.
So notice for example in 12, 26, and 10, 2 is common right, 2 goes into all those. You could rewrite this as 2 times 6x squared minus 13x minus 5. And now what we can do is we can just factor this polynomial. So let's take a look at this polynomial, and at the end we'll bring back the 2. 6x squared minus 13x minus 5, and let's do our little dime now.
So we have to multiply 6 and minus 5, we get minus 30, and then we take the middle term at the bottom which is minus 13, and we want two numbers that multiply, so they should, if we have n and m, n times m should be minus 30 and n plus m should be minus 13. Well, we can try a couple and see. Let's try one and see if it works. 6 times negative 5 is negative 30. Well, but then 6 plus minus 5 is 1. That didn't work out.
So 6 and this, this is wrong. Does not work. D and W.
Does not work. Okay, so scratch that. That didn't work out. Let's try a different one. Well, we know 15 times 2 is 30. So 15, if we tried 15 times minus 2, that's minus 30. 15 plus minus 2, that's positive 13. We're almost there.
Almost. Well, if we just switch the negative sign, we put negative 15 times 2, we get negative 30, and negative 15 plus 2 is minus 13. So this works. So we now know that we need to rewrite this minus 13x as minus 15x plus 2x.
So we go back now and rewrite this as 2 times 6x squared minus 15x. plus 2x minus 5. Okay, great. Now what we're going to do is we're going to factor two terms at a time. So these two terms we're going to factor and then these two terms.
Okay, let's do that. What's in common between 6x squared and 15x? Well, first off the x is definitely in common.
And between 6 and 15, a factor of 3 is also common. So 3x times 2x minus 5, that's what's in common. Now if we look at this guy, the 2x minus 5, there's nothing in common. So if there's nothing in common, you can always pull out a 1, because 1 times the original polynomial itself will give back the original polynomial. So we take out the 1 here, and that's important to write down, so that again, just like last time, we can factor out this 2x minus 5 is in common, 2x minus 5. We can take out the 2x minus 5 and we're left here with 3x plus 1, this 3x and this plus 1. And now we've completed our factorization.
We have 2 times 2x minus 5 times 3x plus 1. And this is the same as the original 12x squared minus 26x minus 10. So this is our final factorization. All right, we're going to stop here. We're just going to do one last bit of factoring in the next video.