### 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, for it to be an identity, the
left-hand side must be the same as the right-hand side. So, what Iβm gonna do is distribute
across the parentheses in our right-hand side to see if it is, in fact, the same as
the left-hand side of our equation. When weβre distributing across
three sets of parentheses, what we do is we deal with two first of all. And then we distribute across the
result and the last parentheses at the end. So, weβre gonna start with π₯ plus
π¦ multiplied by π₯ minus π¦. So, what we want to do is multiply
each term by each other. So, first of all, weβre gonna have
π₯ multiplied by π₯, which gives us π₯ squared. Then, weβre gonna have π₯
multiplied by negative π¦, which gives us negative π₯π¦. And then, we have π¦ multiplied by
π₯, which gives us plus π₯π¦. And then finally, positive π¦
multiplied by negative π¦, which gives us negative π¦ squared.

Well, we just multiplied each of
the terms in the left-hand parentheses by each of the terms in the right-hand
parentheses. But we can use a little memory aid,
and that is FOIL, which means first, so we multiply the first terms together, outer,
multiply the outer terms, then inner, and then last. Okay great, can we simplify
now? Well, we have minus π₯π¦ plus
π₯π¦. So, this is gonna be equal to
zero. So, the result is going to be π₯
squared minus π¦ squared. And we couldβve written this
straightaway from the off because we can see that itβs a formation of the difference
of two squares because we have π₯ and then we have plus and then π¦ and then π₯
minus π¦. Because the signs are both
different and because the final term is the same in both of our parentheses, we
couldβve used the method just to write straightaway π₯ squared minus π¦ squared.

Okay, great. So now, letβs complete the last
part of our distribution across our parentheses. So, what weβre gonna have is π₯
squared minus π¦ squared. Then, this is multiplied by π₯ plus
π¦. So then, weβre gonna have π₯
squared multiplied by π₯, which gives us π₯ cubed. And then, we have π₯ squared
multiplied by positive π¦, which gives us positive π₯ squared π¦. And then, we have negative π¦
squared multiplied by π₯, which gives us negative π₯π¦ squared. And then finally, negative π¦
squared multiplied by positive π¦ will give us negative π¦ cubed. So, this is fully distributed
now.

But this time, the two middle terms
canβt cancel because weβve got π₯ squared π¦ minus π₯π¦ squared. So, we can see that the squareds
are not the same because, in the first term, itβs π₯ squared, and in the second
term, itβs the π¦ thatβs squared. So therefore, we can see that π₯
cubed minus π¦ cubed is not identical to π₯ cubed plus π₯ squared π¦ minus π₯π¦
squared minus π₯ cubed. So therefore, weβve solved the
problem. And we can say that, in answer to
the question, βIs the equation π₯ cubed minus π¦ cubed equal to π₯ plus π¦
multiplied by π₯ minus π¦ multiplied by π₯ plus π¦ an identity?,β the answer is
no.