 Alright, so let's take a look at one of the reasons why it's important to consider what we're actually doing when we do the multiplication. And for this, we'll talk about the multiplication of two polynomials. And the key idea here is that any algebraic multiplication is just an arithmetic operation. So anything that works to multiply two whole numbers can be applied to the multiplication of two polynomials. So we'll focus on just two methods, our partial products method, and our area model. So let's make a comparison here. I'm going to multiply 325 by 46, and I'm going to multiply this. 3x squared plus 2x plus 5 times 4x plus 6. So that 325 times 46. I'll set this up using partial products, so I'll go ahead and split each of these into the different units. And then I'll do my multiplication within each column. So 6 times 5 is 30, 6 times 2 is 12, 6 times 3 is 18, and then 4 times 5 gets me 20, 4 times 2 gets me 8, 4 times 3 gets me 12. And so there I have my different products. And now as long as I stay within each column, I can do the additions. And now I bundle and trade. So this 30 is really three additional tens in the next place over, that gets me 35. This 35 is actually three additional in the next place over. This 29 is two additional in the next place over. This 14 is one in the next place over. And there's my final answer, 14,950. All right, well what if I do the same thing using the polynomial? So again, I'll split the polynomials into the different units. 6 times 5 is 30, 6 times 2x is 12x, 6 times 3x squared is 18x squared, 4x times 5 is 20x, 4x times 2 is 8x squared, 4x times 3 is 12x cubed. And notice that the numbers are the same in both places. 12x cubed to 12, 8, 8x squared, 20x to 20. And so now my final step, I'm going to add them, 30, staying within the columns. And the only real difference is I can't trade for anything in the next place over. The units don't allow me to do that. Well, we could do the same multiplication using the area model. So again, if I wanted to do this multiplication using the area model, I'd break the 325 and the 46 into convenient pieces, 325, 46. And then I'd find the areas of all those rectangles and add them together to find the product. The key here is if we're multiplying the two polynomials, it's exactly the same process. What I can do is I can split this up any way that I want to. I can split this up any way that I want to. And let's do it this way, 3x squared, 2x and 5, 4x and 6. And if I find the areas of each of these individual rectangles, the sum is the product. So the product is add these pieces up, 12x cubed, 26x squared, 32x and 30, and there's my product. And again, the only real difference between the polynomial product and the arithmetic product is I can't combine these in any further capacity. These are just whole numbers so I can add them together. These are different terms and they do not combine.