Polynomial identities introduction | Algebra 2 | Khan Academy

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 1 to that x and then squaring the whole thing.

We saw this 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 more complicated expressions, things that aren't just simple quadratics or that might not be as obvious as this.

The way that we're going to prove whether they're 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. 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 1 is m, m times m is m squared, and then m times m squared is m to the third power. Then I would take this negative 1 and distribute that times every term in that other expression. So, negative 1 times 1 is negative 1. Negative 1 times m is negative m, and negative 1 times m squared is negative m squared.

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 1.

Now, clearly, m to the third power minus 1 is going to be equal to m to the third power minus 1 for any value of m. These are identical expressions, so 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 3 squared plus 2n, is that equal to 8n plus 13? Is this a polynomial identity?” Pause this video and see if you can figure that out.

All right, now we're going to work on that together, and I would do it the exact same way. I would try to simplify with a little bit of algebra. The maybe the easiest thing to do first— and you could do this in multiple ways— is 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 2n from both sides of this equation, I am going to get on the left-hand side n plus 3 squared, and on the right-hand side, I am going to get 6n, 8n minus 2n plus 13.

Now, what's n plus 3 squared? Well, that's going to be n squared plus 2 times 3 times n. 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 6n plus 3 squared, which is 9.

And is this going to be equal to 6n plus 13? Well, already this is starting to look a little bit sketchy, but let's just keep going with the algebra. So, let's see, if we subtract 6n from both sides, what do you get?

Well, on the left-hand side, you're just going to have n squared plus 9, 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 0, this is not going to be true. If n is 1, this is not going to be true. If n is 2, this actually would be true, but if n is 3, this is not going to be true. If n is 4 or 5, etc. 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 the variable in question. So, this one right over here is not a polynomial identity, and we're done.