# Question Video: Proving Polynomial Identities Mathematics

Is the equation π₯Β³ β π¦Β³ = (π₯ β π¦)(π₯Β² + π₯π¦ + π¦Β²) an identity?

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### Video Transcript

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

So, if we look and see whether it is an identity, we wanna see whether the left-hand side is identical to the right-hand side. And to do that, what weβre gonna have to do is distribute across the parentheses. And in order to distribute across our parentheses, what weβre gonna do is multiply each of the terms in the left-hand set by each of the terms in the right-hand set. So first to start with, π₯ multiplied by π₯ squared, which is π₯ cubed. Then, weβre gonna have π₯ multiplied by positive π₯π¦, which is gonna give us π₯ squared π¦. And then, weβre gonna have π₯ multiplied by positive π¦ squared, which is gonna give us π₯π¦ squared.

Well then, next what weβre gonna have is negative π¦ multiplied by π₯ squared, which gives us negative π₯ squared π¦. And then, weβre gonna have negative π¦ multiplied by positive π₯π¦, which is gonna give us negative π₯π¦ squared. And then finally, weβve got negative π¦ multiplied by positive π¦ squared, which is gonna give us negative π¦ cubed.

Okay, great, weβve got to this point, but this doesnβt look like the left-hand side of our equation. But what weβre gonna have to do is tidy up first with some simplifying. Well, first of all, weβve only got one π₯ cubed. Well then, we have π₯ squared π¦ minus π₯ squared π¦, which means these are gonna cancel each other out because theyβre gonna be equal to zero. So then, next weβre gonna have positive π₯π¦ squared minus π₯π¦ squared. So again, these are gonna cancel each other out.

And then finally, weβve got minus π¦ cubed. So, weβre now left with π₯ cubed minus π¦ cubed. Well, this is what we started with on the left-hand side of the equation. So therefore, we can say that the left-hand side and the right-hand side are identical. So therefore, the answer is yes. And we can say that the equation π₯ cubed minus π¦ cubed equals π₯ minus π¦ multiplied by π₯ squared plus π₯π¦ plus π¦ squared is an identity.