# Video: Proving Polynomial Identities

Is the equation 𝑥³ − 𝑦³ = (𝑥 − 𝑦)(𝑥² + 𝑥𝑦 + 𝑦²) an identity?

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### Video Transcript

Is the equation 𝑥 cubed minus 𝑦 cubed equals 𝑥 minus 𝑦 multiplied by 𝑥 squared plus 𝑥𝑦 plus 𝑦 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 we’re gonna have to do is distribute across the parentheses. And in order to distribute across our parentheses, what we’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, 𝑥 multiplied by 𝑥 squared, which is 𝑥 cubed. Then, we’re gonna have 𝑥 multiplied by positive 𝑥𝑦, which is gonna give us 𝑥 squared 𝑦. And then, we’re gonna have 𝑥 multiplied by positive 𝑦 squared, which is gonna give us 𝑥𝑦 squared.

Well then, next what we’re gonna have is negative 𝑦 multiplied by 𝑥 squared, which gives us negative 𝑥 squared 𝑦. And then, we’re gonna have negative 𝑦 multiplied by positive 𝑥𝑦, which is gonna give us negative 𝑥𝑦 squared. And then finally, we’ve got negative 𝑦 multiplied by positive 𝑦 squared, which is gonna give us negative 𝑦 cubed.

Okay, great, we’ve got to this point, but this doesn’t look like the left-hand side of our equation. But what we’re gonna have to do is tidy up first with some simplifying. Well, first of all, we’ve only got one 𝑥 cubed. Well then, we have 𝑥 squared 𝑦 minus 𝑥 squared 𝑦, which means these are gonna cancel each other out because they’re gonna be equal to zero. So then, next we’re gonna have positive 𝑥𝑦 squared minus 𝑥𝑦 squared. So again, these are gonna cancel each other out.

And then finally, we’ve got minus 𝑦 cubed. So, we’re now left with 𝑥 cubed minus 𝑦 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 𝑥 cubed minus 𝑦 cubed equals 𝑥 minus 𝑦 multiplied by 𝑥 squared plus 𝑥𝑦 plus 𝑦 squared is an identity.