WEBVTT
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Is the equation π₯ cubed minus π¦ cubed equals π₯ plus π¦ multiplied by π₯ minus π¦ multiplied by π₯ plus π¦ an identity?
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Well, for it to be an identity, the left-hand side must be the same as the right-hand side.
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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.
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When weβre distributing across three sets of parentheses, what we do is we deal with two first of all.
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And then we distribute across the result and the last parentheses at the end.
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So, weβre gonna start with π₯ plus π¦ multiplied by π₯ minus π¦.
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So, what we want to do is multiply each term by each other.
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So, first of all, weβre gonna have π₯ multiplied by π₯, which gives us π₯ squared.
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Then, weβre gonna have π₯ multiplied by negative π¦, which gives us negative π₯π¦.
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And then, we have π¦ multiplied by π₯, which gives us plus π₯π¦.
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And then finally, positive π¦ multiplied by negative π¦, which gives us negative π¦ squared.
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Well, we just multiplied each of the terms in the left-hand parentheses by each of the terms in the right-hand parentheses.
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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.
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Okay great, can we simplify now?
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Well, we have minus π₯π¦ plus π₯π¦.
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So, this is gonna be equal to zero.
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So, the result is going to be π₯ squared minus π¦ squared.
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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¦.
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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.
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Okay, great.
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So now, letβs complete the last part of our distribution across our parentheses.
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So, what weβre gonna have is π₯ squared minus π¦ squared.
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Then, this is multiplied by π₯ plus π¦.
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So then, weβre gonna have π₯ squared multiplied by π₯, which gives us π₯ cubed.
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And then, we have π₯ squared multiplied by positive π¦, which gives us positive π₯ squared π¦.
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And then, we have negative π¦ squared multiplied by π₯, which gives us negative π₯π¦ squared.
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And then finally, negative π¦ squared multiplied by positive π¦ will give us negative π¦ cubed.
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So, this is fully distributed now.
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But this time, the two middle terms canβt cancel because weβve got π₯ squared π¦ minus π₯π¦ squared.
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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.
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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.
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So therefore, weβve solved the problem.
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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.