Polynomial identities introduction

- [Instructor] What we're going to do in this video is talk a little bit about polynomial identities, and this is really just a fancy way of seeing whether an expression that involves a polynomial is equal to another expression. So for example, you're familiar with x squared plus two x plus one, we've seen polynomials like this multiple times, this is a quadratic, and you might recognize that this would be equal to x plus one squared, that for any value of x, x squared plus two x plus one is the same thing as adding one to that x, and then squaring the whole thing. And we saw this when we first got, when we first learned how to multiply binomials and we took the square of binomials, but now we're going to do this with slightly complicated expressions, things that aren't just simple quadratics or that might not be as obvious as this. And the way that we're gonna prove whether they are true or not is just with a little bit of algebraic manipulation. So for example, if someone walked up to you on the street and said, all right, m to the third minus one, is it equal to m minus one times one plus m plus m squared? Pause this video and see what you would tell that person, whether you could prove whether it is or is not a true polynomial identity. Okay, let's do it together, and the way I would tackle this is I would expand out, I would multiply out what we have on the right-hand side, so this is going to be equal to, so first, I could take this m and then multiply it times every term in this second expression, so m times one is m, m times m is m squared, and then m times m squared is m to the third power, and then I would take this negative one, and then multiply and then distribute that times every term in that other expression, so negative one times one is negative one, negative one times m is negative m, and negative one times m squared is negative m squared, and now let's see if we can simplify this. We have an m and a negative m, so those are going to cancel out, we have an m squared and a negative m squared, so those cancel out, and so we are going to be left with m to the third power, minus one. Now, clearly, m to the third power minus one is going to be equal to m to the third power minus one for any value of m, these are identical expressions. So this is, this is indeed a polynomial identity. Let's do another example. Let's say someone were to walk up to you on the street and said, quick, n plus three squared plus two n, is that equal to eight n plus 13, is this a polynomial identity? Pause this video and see if you can figure that out. All right, now we're gonna work on that together. And I would do it the exact same way. I would try to simplify with a little bit of algebra, maybe the easiest thing to do first, and you could do this in multiple ways is, I have this, I have these n terms, two n's here, eight n's over here, well, what if I were to get these two n's out of the left-hand side, so if I were to just subtract two n from both sides of this equation, I'm going to get on the left-hand side, n plus three squared, and on the right-hand side, I'm going to get six n, eight n minus two n, plus 13. Now, what's n plus three squared? Well, that's going to be n squared plus two times three times n. And if what I just did does not seem familiar to you, I encourage you to look at the videos about squaring binomials, but this is going to be plus six n, plus three squared, which is nine, and is this going to be equal to six n plus 13? Well, already this is starting to look a little bit, a little bit sketchy, but let's just keep going with the algebra. So let's see if we subtract six n from both sides, what do you get? Well, on the left-hand side, you're just going to have n squared plus nine, and on the right-hand side, you're going to get 13. Now, are there values of n for which this is not always true? Well, sure. I can find a lot of values of n for which this is not always true. If n is a zero, this is not going to be true. If n is one, this is not going to be true. If n is two, this actually would be true, but if n is three, this is not going to be true. If n is four, five, et cetera, so for actually most values of n, this is not going to be true. So in order for it to be a polynomial identity, it has to be true for all of the values that are legitimate values that you can evaluate for those, for the variable in question. So this one right over here is not a polynomial, polynomial identity, and we are done.