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.
