 So let's take a look at what happens when you multiply two polynomials. So again, the reason that basic arithmetic is important is not so much because you need to be able to do basic arithmetic, but because it paves the path for learning algebra. And in this particular case, the understanding of multiplication as a repeated sum, along with the understanding of a number as a sum of terms, allows us to extend arithmetic to including algebraic expressions. On the other hand, if you just know how to push digits around and know quote-unquote how to do basic arithmetic, you will find that algebra requires learning an entirely new set of rules instead of being a natural extension of arithmetic. So for example, let's consider the product 27 times 31 with the product 2x plus 7 times 3x plus 1. And so let's take a look at that, that's 31 times 27. And if I applied the partial products method, what I'm actually finding here is I'm finding the product of 27, 30, and 1, that's 20 times 30, that's 20 times 1, 7 times 30, and 7 times 1, and those are my partial products, and my product is going to be the sum of all those individual terms. On the other hand, if I take a look at the product 3x plus 1 times 2x plus 7, well again, here I reconsidered this to be 30 and 1, 20 and 7, and then I found the product of every term with every other term. Here this is 2x and 7, 3x and 1, so I'm going to find the product of every term with every term. So what am I going to do? Well 2x times 3x, well that's this 20 times 30, 2x times 3x is 6x squared, 20 times 1 over here, 2x times 1, 2x, 7 times 30, 7 times 30, 7 times 3x, 21x, 7 times 1, 7 times 1 is going to be 7, and our partial products are here, and our final product is just going to be the sum of all of these things put together. So there's my product. Probably at some point, depending on how you learned algebra, you learned this method of multiplying two things that looked exactly like this using some silly little acronym like FIAL or something like that, but that's great for multiplying something that looks exactly like this, but what if you have something that looks like this? Well you can't apply FIAL to that and get the correct answer. On the other hand, this is really multiplying 230 and 7 by 150 and 4, and if you understand how you multiply 230 and 7 by 154, then you understand immediately how to multiply these two things together. Let's go ahead and do that. So again, I'll set down our factors, and if I think about this in terms of partial products, well what would I do? I'd multiply 100 times 200, 100 times 30, 100 times 7, 50 times 250 times 30, 50 times 7, and so on, and I'm going to do exactly the same thing. So my x squared times 2x squared, 2x to the fourth, and as with partial products, I want to write down what these are in the correct place value. There really isn't a place of value corresponding to polynomials, but I might keep the terms of the same degree in separate locations on the page just so that I can keep track of them. Again, arithmetic is bookkeeping, and the same is true for algebra. So let's see x squared times 2x squared, that's 2x to the fourth, I'll write it down here, x squared times 3x is 3x to the third. It's not an x to the fourth, so I'll put that in a new column. X squared times 7 is 7x squared, again new term, so now I have my terms fourth degree, third degree, second degree terms. And now I'm going to continue, 5x times 2x squared is 10x to the third, that should go in this column, 5x times 3x is 15x squared, I'll put that in this column, 5x times 7 is 35x, I'll start a new column for those, and now 4 times 2x squared, well that's going to be 8x squared, 4 times 3x, that's a 12x, goes here, and 4 times 7 is 28, start a new column for that product, and again these are my partial products, so the actual product is going to be the sum of everything I have, 2x to the fourth, 13x cubed, 30x squared, 47x and 28, and there's my product.