Video Transcript
Factor four 𝑥 squared times 𝑥
squared minus six 𝑦 squared plus nine 𝑦 to the power of four by completing the
square.
Let’s start by multiplying out the
first term to see what we’re working with. Notice that we have a pair of
perfect squares here. Recall the standard expansion of
the binomial 𝑎 plus 𝑏 squared as 𝑎 squared plus two 𝑎𝑏 plus 𝑏 squared. This yields the equation 𝑎 squared
plus 𝑏 squared equals 𝑎 plus 𝑏 squared minus two 𝑎𝑏.
We’re going to set 𝑎 squared to
four 𝑥 to the four and 𝑏 squared to nine 𝑦 to the four. This gives 𝑎 equals two 𝑥 squared
and 𝑏 equals three 𝑦 squared. Multiplying 𝑎 and 𝑏 together and
then by two, we have two 𝑎𝑏 equals 12𝑥 squared 𝑦 squared. Using the starred identity, we have
four 𝑥 to the power four plus nine 𝑦 to the power four equals two 𝑥 squared plus
three 𝑦 squared all squared minus 12𝑥 squared 𝑦 squared.
We can now make this substitution
in our expression. The sum of four 𝑥 to the fourth
and nine 𝑦 to the fourth becomes two 𝑥 squared plus three 𝑦 squared all squared
minus 12𝑥 squared 𝑦 squared. And the 24𝑥 squared 𝑦 squared
term carries over. Collecting like terms, we have two
𝑥 squared plus three 𝑦 squared all squared minus 36𝑥 squared 𝑦 squared.
Notice that this second term, 36𝑥
squared 𝑦 squared, is itself a perfect square. We have transformed our original
expression into a difference of two squares. We factor this in the normal way as
the product of the sum and the difference. And this is as far as we can factor
this expression over the rationals. As an aside, observe that if we
allow irrational coefficients, we can actually break it into linear factors using
quadratic formula.
