Video: Proving Polynomial Identities

Is the equation π‘₯Β³ βˆ’ 𝑦³ = (π‘₯ + 𝑦)(π‘₯ βˆ’ 𝑦)(π‘₯ + 𝑦) an identity?

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Video Transcript

Is the equation π‘₯ cubed minus 𝑦 cubed equals π‘₯ plus 𝑦 multiplied by π‘₯ minus 𝑦 multiplied by π‘₯ plus 𝑦 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 we’re gonna do is doing it in stages. So first of all, I’m 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, we’ve got π‘₯ multiplied by negative 𝑦, which gives us negative π‘₯𝑦. And then, we’ve got 𝑦 multiplied by π‘₯. So we’re gonna get plus π‘₯𝑦. And then, finally, we’ve got positive 𝑦 multiplied by negative 𝑦 which is gonna give us negative 𝑦 squared. Okay, so now, we’ve got π‘₯ squared minus π‘₯𝑦 plus π‘₯𝑦 minus 𝑦 squared.

Well, if we’ve got negative π‘₯𝑦 plus π‘₯𝑦, well, this means we’re gonna cancel each other out because we’re gonna get zero. Well, fully distributed, the first two parentheses are gonna give us π‘₯ squared minus 𝑦 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 that’s because what we have here is a difference of two squares. So now, what we’re gonna do is bring down the π‘₯ plus 𝑦, so the last pair of parentheses.

So now, we’ve got π‘₯ squared minus 𝑦 squared multiplied by π‘₯ plus 𝑦. And what we want to do is see whether this will in fact give us our π‘₯ cubed minus 𝑦 cubed. But what we’re 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, we’ve got positive π‘₯ squared multiplied by positive 𝑦, which gives us plus π‘₯ squared 𝑦. Then, we’re gonna get minus π‘₯ squared 𝑦. So once again, we can see that we’re gonna have two terms that can cancel. That’s cause we’ve got positive π‘₯ squared 𝑦 and negative π‘₯ squared 𝑦. So these cancel each other out. So therefore, we’re left with π‘₯ cubed minus 𝑦 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 we’ve got as an identity. So we’d have π‘₯ cubed minus 𝑦 cubed is and then identical to π‘₯ plus 𝑦 multiplied by π‘₯ minus 𝑦 multiplied by π‘₯ plus 𝑦. And that’s what these three horizontal lines mean. That means is an identity or identical to. So the correct answer to the question is yes.

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