Video Transcript
Factor 𝑥 to the eighth power minus
16𝑦 to the eighth power by completing the square.
In this question, it may not
immediately be obvious how we can solve the problem by completing the square. However, we do notice that our
expression is of the form 𝑎 squared minus 𝑏 squared. This is known as the difference of
squares and can be factored as 𝑎 plus 𝑏 multiplied by 𝑎 minus 𝑏. Letting 𝑎 squared equal 𝑥 to the
eighth power, we know that 𝑎 is equal to 𝑥 to the fourth power. And letting 𝑏 squared equals 16𝑦
to the eighth power, we know that 𝑏 is the square root of this, which is equal to
four 𝑦 to the fourth power. The original expression can
therefore be rewritten as 𝑥 to the fourth power all squared minus four 𝑦 to the
fourth power all squared. And factoring this using the
difference of squares, we have 𝑥 to the fourth power plus four 𝑦 to the fourth
power multiplied by 𝑥 to the fourth power minus four 𝑦 to the fourth power.
The second part of our expression
is once again written in the form 𝑎 squared minus 𝑏 squared. 𝑥 to the fourth power minus four
𝑦 to the fourth power is equal to 𝑥 squared plus two 𝑦 squared multiplied by 𝑥
squared minus two 𝑦 squared. These two parentheses cannot be
factored any further. So we now need to consider the
expression 𝑥 to the fourth power plus four 𝑦 to the fourth power. It is this expression that we’ll be
able to factor by completing the square. This is in the form 𝑐 squared plus
𝑑 squared. And we need to manipulate this so
it is in the form of a perfect square trinomial.
We know that any perfect square
trinomial of the form 𝑐 squared plus two 𝑐𝑑 plus 𝑑 squared can be factored into
the form 𝑐 plus 𝑑 all squared. Since 𝑐 squared is equal to 𝑥 to
the fourth power, 𝑐 is equal to 𝑥 squared. Likewise, since 𝑑 squared is equal
to four 𝑦 to the fourth power, 𝑑 is equal to two 𝑦 squared. The term two 𝑐𝑑 is therefore
equal to two multiplied by 𝑥 squared multiplied by two 𝑦 squared. And this is equal to four 𝑥
squared 𝑦 squared.
We need to add the positive and
negative of this to the expression in our first set of parentheses to create our
perfect square trinomial. The full expression can be written
as shown. And we can now factor the perfect
square trinomial contained in the first set of parentheses. This is equal to 𝑥 squared plus
two 𝑦 squared all squared. And rewriting four 𝑥 squared 𝑦
squared as two 𝑥𝑦 all squared, we have 𝑥 squared plus two 𝑦 squared all squared
minus two 𝑥𝑦 all squared multiplied by 𝑥 squared plus two 𝑦 squared multiplied
by 𝑥 squared minus two 𝑦 squared.
We note that the expression within
the brackets is the difference of squares. And using the rule we saw earlier,
this factors to 𝑥 squared plus two 𝑦 squared plus two 𝑥𝑦 multiplied by 𝑥
squared plus two 𝑦 squared minus two 𝑥𝑦. We can then reorder the terms in
these parentheses as shown, giving us a final answer of 𝑥 squared plus two 𝑥𝑦
plus two 𝑦 squared multiplied by 𝑥 squared minus two 𝑥𝑦 plus two 𝑦 squared
multiplied by 𝑥 squared plus two 𝑦 squared multiplied by 𝑥 squared minus two 𝑦
squared. This is the fully factored form of
𝑥 to the eighth power minus 16𝑦 to the eighth power.
