 So let's round off our discussion of factoring by talking about sweeping things under the rug. And what we mean by that is the following. A useful skill in mathematics and life is to recognize the form of an expression. So 3 plus 5 and 3,419,874 plus 15,932 have the same form. They're both A plus B. Likewise, the equation 5x plus 3 equals 12 and the equation 317's y plus square root of pi equals 5th root of 3 have the same form. They're both equations of the form Ax plus B equals C. And x to the 10th minus 5 and z squared minus 1 have the same form. They're both A squared minus B squared. And the reason that it's useful to recognize these have the same form is that problems with the same form are solved in the same way. So let's say we want to factor this expression. So the thing to recognize here is that we have this 3x plus 5 showing up in a couple of places. And so if we sweep this under the rug, ignore it, then what we end up with looks like a normal quadratic polynomial. And so that suggests since both terms have a 3x plus 5 in them, we can replace 3x plus 5 with some variable. So in a spasm of inspired creativity, we'll use y equals 3x plus 5. Remember, equals means interchangeable, so every time we see a 3x plus 5, we'll replace it with y. And now we have something a lot more familiar, y squared minus 4y minus 12, so we'll factor. And at this point, we need to invoke a rule that you learned in kindergarten. Put things back where you found them. In particular, since we started with the problem of factoring an expression in x, we want to make sure that when we factor, we have an expression in x. So again, equals means replaceable. So every time we see a y, we can replace it with 3x plus 5. And finally, let's do a little cleanup of the terms inside the parentheses because we can, which gives us our factorization. Or let's try to factor x to the eighth minus 16. The important thing to remember is there is no substitute for careful observation. You must look and analyze. So here, maybe one of the things we'll notice is that x to the eighth is x to the fourth squared, and 16 is 4 squared. And so x to the eighth minus 16 is x to the fourth squared minus 4 squared. And if we sweep the items inside the parentheses under the rug, we see that we actually have a difference of perfect squares, and we know how to factor that. So we can factor, and so we have our factorization. Or do we? So the important thing to remember here is don't stop until you're done. Again, there's no substitute for careful observation. And if we take a look at these two factors, we notice that x to the fourth is x squared squared, and 4 is 2 squared. So that means x to the fourth minus 4 is x squared squared minus 2 squared, making this a difference of perfect squares, and we know how to factor that. So we can factor x to the fourth minus 4, and a large part of success in algebra and in life is your ability to do bookkeeping. We haven't done anything to the x to the fourth plus 4. It's still there, so we should write it down. Could we do something with it? Unfortunately, this is a sum of perfect squares, and we don't have a factorization for that. So I think we're going to have to leave that. On the other hand, we might try to go a little bit further with these other factors. We observe that x squared is x squared, and 2 is, well, that's square root of 2 squared. So x squared minus 2 is x squared minus square root of 2 squared, making this a difference of perfect squares. Well, not quite. Square root of 2 is not a perfect square, and our corresponding factorization would be x minus square root of 2 times x plus square root of 2, which are not rational factors. Since we're primarily interested in finding rational factors, we don't take this step for now. And the important idea to keep in mind is we could, and it may be useful to do so in other courses. But for now, we can't factor this any further.