### Video Transcript

Is the equation π₯ cubed minus π¦ cubed equals π₯ plus π¦ multiplied by π₯ minus π¦ multiplied by π₯ plus π¦ an identity?

Well, first of all, what do we mean by an identity? Well, an identity means that the left-hand side is in fact the same as the right-hand side. So we can work that out by distributing across our parentheses. But in order to do this, what weβre gonna do is doing it in stages. So first of all, Iβm gonna distribute across the first two sets of parentheses. So first of all, we have π₯ multiplied by π₯, which gives us π₯ squared. Then, weβve got π₯ multiplied by negative π¦, which gives us negative π₯π¦. And then, weβve got π¦ multiplied by π₯. So weβre gonna get plus π₯π¦. And then, finally, weβve got positive π¦ multiplied by negative π¦ which is gonna give us negative π¦ squared. Okay, so now, weβve got π₯ squared minus π₯π¦ plus π₯π¦ minus π¦ squared.

Well, if weβve got negative π₯π¦ plus π₯π¦, well, this means weβre gonna cancel each other out because weβre gonna get zero. Well, fully distributed, the first two parentheses are gonna give us π₯ squared minus π¦ squared. Now, we work out by distributing across our parentheses. But we could have done that straightaway. And we could have known that that was what the answer was going to be. And thatβs because what we have here is a difference of two squares. So now, what weβre gonna do is bring down the π₯ plus π¦, so the last pair of parentheses.

So now, weβve got π₯ squared minus π¦ squared multiplied by π₯ plus π¦. And what we want to do is see whether this will in fact give us our π₯ cubed minus π¦ cubed. But what weβre going to do now is distribute across these parentheses. Well, first of all, we get π₯ squared multiplied by π₯ which is gonna give us π₯ cubed. Then, weβve got positive π₯ squared multiplied by positive π¦, which gives us plus π₯ squared π¦. Then, weβre gonna get minus π₯ squared π¦. So once again, we can see that weβre gonna have two terms that can cancel. Thatβs cause weβve got positive π₯ squared π¦ and negative π₯ squared π¦. So these cancel each other out. So therefore, weβre left with π₯ cubed minus π¦ cubed. So we can see that the left-hand side of the equation is the same as the right-hand side of the equation.

So then, what we could do is rewrite what weβve got as an identity. So weβd have π₯ cubed minus π¦ cubed is and then identical to π₯ plus π¦ multiplied by π₯ minus π¦ multiplied by π₯ plus π¦. And thatβs what these three horizontal lines mean. That means is an identity or identical to. So the correct answer to the question is yes.