Video Transcript
Factor 16𝑥 to the fourth power minus 25𝑥 squared 𝑦 squared plus nine 𝑦 to the
fourth power fully by completing the square.
We want to factor this expression by completing the square. So we need to manipulate it 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 16𝑥 to the fourth power and 𝑏 squared
to be nine 𝑦 to the fourth power, then our value of 𝑎 is the square root of 𝑎
squared, which is equal to four 𝑥 squared. And our value of 𝑏 is the square root of 𝑏 squared, which is equal to three 𝑦
squared. Then, our middle term is equal to two 𝑎𝑏, or sometimes negative two 𝑎𝑏. Two 𝑎𝑏 comes out to two times four 𝑥 squared times three 𝑦 squared, which is 24𝑥
squared 𝑦 squared.
In our next step, we will introduce the two 𝑎𝑏 term into the original expression,
while also moving the negative 25𝑥 squared 𝑦 squared term to the end. 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 24𝑥 squared 𝑦
squared minus 24𝑥 squared 𝑦 squared. Our expression with these new terms is 16𝑥 to the fourth power plus 24𝑥 squared 𝑦
squared plus nine 𝑦 to the fourth power minus 25𝑥 squared 𝑦 squared minus 24𝑥
squared 𝑦 squared.
We can now factor the first three terms as a perfect square trinomial, giving us four
𝑥 squared plus three 𝑦 squared squared. Then, we can combine the like terms, giving us negative 49𝑥 squared 𝑦 squared. We now have a difference of squares, since the expression within the parentheses is
being squared and 49𝑥 squared 𝑦 squared is a perfect square, specifically, the
square of seven 𝑥𝑦, where 𝑎 is in the first parentheses and 𝑏 is in the second
parentheses. Following the formula for factoring a difference of two squares, we get four 𝑥
squared plus three 𝑦 squared minus seven 𝑥𝑦 times four 𝑥 squared plus three 𝑦
squared plus seven 𝑥𝑦.
We then check whether the resulting polynomials within each set of parentheses can be
factored. In this case, neither polynomial is fully factored yet. Using factoring by grouping, we find the full factorization of the first polynomial
is four 𝑥 minus three 𝑦 times 𝑥 minus 𝑦. And the full factorization of the second polynomial is four 𝑥 plus three 𝑦 times 𝑥
plus 𝑦.
Therefore, we have that four 𝑥 plus three 𝑦 times four 𝑥 minus three 𝑦 times 𝑥
plus 𝑦 times 𝑥 minus 𝑦 represents the full factorization of 16𝑥 to the fourth
power minus 25𝑥 squared 𝑦 squared plus nine 𝑦 to the fourth power.
