### Video Transcript

Factorize fully nine 𝑥 to the
fourth power minus 21𝑥 squared 𝑦 squared plus four 𝑦 to the fourth power.

We notice that the polynomial
contains two perfect square terms: nine 𝑥 to the fourth power and four 𝑦 to the
fourth power. So we will attempt to factor this
expression by completing the square. To use this method, we need to
manipulate the expression to include a perfect square trinomial in the form 𝑎
squared plus or minus two 𝑎𝑏 plus 𝑏 squared, which can be factored as 𝑎 plus or
minus 𝑏 squared. In these trinomials, 𝑎 and 𝑏 may
be variables, constants, or products of variables and constants.

In this example, if we take 𝑎
squared to be nine 𝑥 to the fourth power and 𝑏 squared to be four 𝑦 to the fourth
power, then our value of 𝑎 is the square root of 𝑎 squared, which is equal to
three 𝑥 squared. And our value of 𝑏 is the square
root of 𝑏 squared, which is equal to two 𝑦 squared. Then, our middle term is equal to
two 𝑎𝑏, or in some cases negative two 𝑎𝑏. Two 𝑎𝑏 comes out to two times
three 𝑥 squared times two 𝑦 squared, which is 12𝑥 squared 𝑦 squared.

In our next step, we will introduce
the two 𝑎𝑏 term into the original expression, while also moving the negative 21𝑥
squared 𝑦 squared term to the end of the expression. For any term we introduce into the
expression, we must add the same term with the opposite sign. This way, we are effectively adding
zero, which does not change the polynomial. In this case, the zero gets added
to the polynomial in the form of 12𝑥 squared 𝑦 squared minus 12𝑥 squared 𝑦
squared.

Our expression with these new terms
is nine 𝑥 to the fourth power plus 12𝑥 squared 𝑦 squared plus four 𝑦 to the
fourth power minus 21𝑥 squared 𝑦 squared minus 12𝑥 squared 𝑦 squared.

We can now factor the first three
terms as a perfect square trinomial, giving us three 𝑥 squared plus two 𝑦 squared
squared. Then, we can combine the like
terms, giving us negative 33𝑥 squared 𝑦 squared. Now at this step, we expect to have
a difference of squares. However, 33 is not a perfect
square. When this happens, we can try
forming a perfect square trinomial with a negative two 𝑎𝑏 term instead of the
positive two 𝑎𝑏 term we chose. So we will erase a bit of our work
to return to the previous step.

We proceed to change the sign in
front of the two 𝑎𝑏 term to a negative. This means our new expression is
nine 𝑥 to the fourth power minus 12𝑥 squared 𝑦 squared plus four 𝑦 to the fourth
power minus 21𝑥 squared 𝑦 squared plus 12𝑥 squared 𝑦 squared. Then, we again factor the first
three terms as a perfect square trinomial, but this time the factors are of the form
𝑎 minus 𝑏.

Now when we combine the like terms,
we have negative nine 𝑥 squared 𝑦 squared. This is a difference of squares
since the expression within the parentheses is being squared and nine 𝑥 squared 𝑦
squared is a perfect square, specifically the square of three 𝑥𝑦, where 𝑎 is in
the first parentheses and 𝑏 is in the second parentheses.

Following the formula for factoring
a difference of squares, we get three 𝑥 squared minus two 𝑦 squared minus three
𝑥𝑦 times three 𝑥 squared minus two 𝑦 squared plus three 𝑥𝑦.

Finally, we need to check whether
the resulting polynomials within each set of parentheses can be factored. In this case, both polynomials are
prime. So we have that three 𝑥 squared
minus three 𝑥𝑦 minus two 𝑦 squared times three 𝑥 squared plus three 𝑥𝑦 minus
two 𝑦 squared represents the full factorization of nine 𝑥 to the fourth power
minus 21𝑥 squared 𝑦 squared plus four 𝑦 to the fourth power.