Video Transcript
Is the equation 𝑥 to the fourth
power minus 𝑦 to the fourth power over 𝑥 squared minus 𝑦 squared equals 𝑥
squared plus 𝑦 squared an identity?
Remember, an identity is an
equation that holds for all values of 𝑥 or all values of the variables. In this case, we need to decide
whether this equation holds for all values of 𝑥 and 𝑦. Now, to prove an equation is an
identity, it’s simply not enough to substitute a few values of 𝑥 and 𝑦 in and
check that it works for those values. Instead, we’re going to start with
the expression 𝑥 to the fourth power minus 𝑦 to the fourth power over 𝑥 squared
minus 𝑦 squared and see how we might manipulate that to look more like a standard
polynomial. Now, the key to simplifying here is
to spot that we have a quotient of two expressions that are the difference of two
squares.
Remember, 𝑎 squared minus 𝑏
squared can be factored as 𝑎 plus 𝑏 times 𝑎 minus 𝑏. Now what this means is we can
factor or factorize both parts, the numerator and the denominator. Now, in fact, we’re just going to
factor the numerator, and we’ll see why in a moment. Comparing this expression to the
general form, we’re going to let 𝑎 be equal to 𝑥 squared and 𝑏 be equal to 𝑦
squared. And this is because 𝑥 squared
squared gives us 𝑥 to the fourth power. And we can therefore factor 𝑥 to
the fourth power minus 𝑦 to the fourth power as 𝑥 squared plus 𝑦 squared times 𝑥
squared minus 𝑦 squared.
Now, what do we notice? There is a common factor on both
the numerator and denominator of our fraction. And that common factor is 𝑥
squared minus 𝑦 squared. We’re therefore going to divide
through by 𝑥 squared minus 𝑦 squared. On the numerator, that leaves us
with 𝑥 squared plus 𝑦 squared. And on the denominator, we get
one. So this simplifies simply to 𝑥
squared plus 𝑦 squared. We can therefore say that for all
values of 𝑥, 𝑥 to the fourth power minus 𝑦 to the fourth power over 𝑥 squared
minus 𝑦 squared is equal to 𝑥 squared plus 𝑦 squared. Since this is true for all values
of 𝑥, we know we have an identity. And so the answer is yes. This equation is indeed an
identity.
