 In this video, I'll discuss how we can build a polynomial ring. So let's start with an example we're already familiar with. So we're going to use the symbol r bracket x to denote the set of all polynomials with real coefficients. Now this forms a ring where addition is the usual polynomial addition that is we combine like terms. So if you had something like say like x squared plus 3x plus 7 and you add this to like say x cubed minus 2x squared plus 5, again, you just combine like terms, in which case you would get x cubed minus x squared plus 3x plus 12. That would be the sum. We just combine like terms. Then multiplication of polynomials is going to be defined by the usual sort of like foil method. You just use the distributive property, which case if we had something like x squared plus 2x plus 1 and we times that by say x cubed minus 5x, the usual multiplication here, the usual like distributive property, which is the extended foil method, you're going to get x squared times x cubed versus x to the fifth x squared minus 5x. It's going to give you a minus, we're going to put some little space here, we're going to get a minus 5x cubed right there. Then the next one you're going to take a 2x times a 3x, which gives us a 2x to the fourth. You're going to take a 2x times a negative 5x, which gives us a negative 10x squared. And then lastly, you're going to get a plus 1 times x cubed, which gives us an x cubed. And then you're going to get a minus 5x, for which then if there are any like terms, you would combine them together. So the product would be x to the fifth plus 2x to the fourth minus 4x cubed. There were some like terms there minus 10x squared minus 5x. That would be the polynomial product for which this thing is just the usual, the usual multiplication and addition of polynomials. This forms a ring. And in fact, rx here will form a commutative ring with unity, multiplication of polynomials does commute. And the unity will just be the constant number one, rx does include constant polynomials in this situation. Now we can generalize this construction. So in fact, if r is any ring whatsoever, we can define the polynomial ring, which will be denoted r of x. So r of x will be the set of all polynomials whose coefficients come from r. And so how do we add together polynomials? Well, we're just going to take the usual, let's take the usual combined like terms approach to add polynomials. So I had something like ax squared plus bx plus c. And then I want to add this to let's say like dx cubed plus ex squared plus f or whatever something like that. How do you add like terms? Well, you look for the x cubed terms, there's just a dx cubed. The x squared terms, there are two x squares in play here. So you're going to take a plus e x squared, you're going to get a bx and then there are two constants in play here, you have c and f right here, you get c plus f, where in this situation, we're assuming a b c d e and f all belong to our ring. So when you combine like terms, the only thing that matters is that you can add together the coefficients. And this is the right way to approach it here because if you had something like ax squared plus ex squared, this is how addition ought to be defined because if I'm making a ring of polynomials, that ring should be distributive so I can factor out the x squared, in which case you get a plus e times x squared. This is the only way that multiplication could be defined using like terms. So if you can add together the coefficients, then you can add polynomials, that would have to be the case. What about multiplication? Well, multiplication is going to work the same way. If we had something like let's say like a one x squared plus b one x plus c one, and we're going to multiply that by like say a one x cubed plus b sorry a two x cubed plus b two x something like that. When you foil that out, then what you're going to get here is by the distributive law, right? By the usual distributive law, you're going to get something like the following. You're going to get an a one x two times a two x three plus b two x, you're going to get b one x times a two x three plus b two x, and then you're going to get a c one times a two x cubed plus b two x. So the distributive law was originally defined for just two elements in the sum. The distributive law can be extended to any arbitrary sum, which follows very quickly by induction here, for which then you're going to use the distributive property again after all the foil method basically means you distribute it twice with the two binomials. So we're going to distribute this thing again. And in case we end up with an a one x two times an a two x cubed like so plus a one x two times b two x, then the next one would give us, we're going to get some b one x times a two x three plus we're going to get some b one x times b two x like so. And then lastly, we'll get some c one times a two x cubed. And then we end up with some of the last one there c one times b two x. So again, respecting the order of the product right here, you then multiply these things out, multiply the coefficients together, you're going to get an a one a two x to the fifth. Next you'll get an a one b two x cubed. Next we get a b one a two a two times x to the fourth. Then you'll get a b one b two x squared. The next one will end up with a c one a two x cubed. And then lastly, we'll get a c one b two times x, for which if there's any like terms, we could add them together. So we have an x cubed and x cubed, that's a like, in which case, then if we put all these together, we get an a one a two times x to the fifth. Then we get a b one a two x to the fourth for the a three x three excuse me, we're going to get a one b two plus a c one a two x cubed. We had the b one b two x squared, and then a c one b two times x. So you can see this example that in order to multiply out the polynomial, what we needed is we need to have, we need multiplication, of course, of the coefficients. We also need addition of the coefficients. Basically, we need a ring in order to work with polynomials. And so therefore this forms the so-called polynomial ring. Okay, for which to form this polynomial ring, if our coefficients form a ring, then we can form this ring of polynomials, for which the usual addition of polynomials and usual multiplication of polynomials are the operations in play. This will form a ring, which for which addition is going to form will be a associative commutative. It has an identity, which is just a zero polynomial. It'll have inverses with just the negative polynomial. Multiplication will be associative. Multiplication will be distributive. Those are things we can guarantee for this ring. Now, if the ring of coefficients is commutative, then the polynomial ring will be commutative as well. So rx is commutative if and only if r is commutative. The polynomial ring will be commutative if and only if the coefficient ring is commutative. And notice here that I was very careful when I did my products. I never twisted the order. It's always a first term, second term, first term, second term, first term, second term, first term, second term, first term, second term, first term, second term. You see that. So even if the ring is non-commutative, we can make sense out of this polynomial multiplication. But if the multiplication of the coefficients is commutative, then the multiplication of the whole ring will be commutative. This is also true for rings with unity. r of x, the polynomial ring, will have unity if and only if r is a ring with unity, for which the unity will just be the constant polynomial one. So you'll get that f of x equals the constant polynomial one. That would be the unity of these rings. And so given a ring, it could be a commutative ring, commutative ring with unity, just ring with unity or just a ring, we can construct the so-called polynomial ring denoted r of x.