Video Transcript
Factor 16𝑥 to the fourth power 𝑦 squared plus four 𝑦 squared 𝑧 to the fourth
power by completing the square.
To begin, we will consider whether the two terms in the given polynomial have a
highest common factor, or HCF, which may contain variables, constants, or products
of variables and constants.
We determine that four 𝑦 squared is the highest common factor of the two terms. By dividing each term by the HCF, we find the remaining terms in the parentheses to
be four 𝑥 to the fourth power plus 𝑧 to the fourth power. 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 four 𝑥 to the fourth power and 𝑏
squared to be 𝑧 to the fourth power, then our value of 𝑎 is the square root of 𝑎
squared, which is equal to two 𝑥 squared. And our value of 𝑏 is the square root of 𝑏 squared, which is equal to 𝑧
squared. Then, our middle term is equal to two 𝑎𝑏, or in some cases negative two 𝑎𝑏. Two 𝑎𝑏 comes out to two times two 𝑥 squared times 𝑧 squared, which is four 𝑥
squared 𝑧 squared.
In our next step, we will introduce the two 𝑎𝑏 term into the original
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 four 𝑥 squared 𝑧
squared minus four 𝑥 squared 𝑧 squared. Our expression with these new terms is four 𝑦 squared times four 𝑥 to the fourth
power plus four 𝑥 squared 𝑧 squared plus 𝑧 to the fourth power minus four 𝑥
squared 𝑧 squared. We can now factor the first three terms in the parentheses as a perfect square
trinomial, giving us two 𝑥 squared plus 𝑧 squared squared.
Now we have a difference of squares, since the expression within the parentheses is
being squared and four 𝑥 squared 𝑧 squared is a perfect square, specifically the
square of two 𝑥𝑧, 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 two 𝑥 squared
plus 𝑧 squared minus two 𝑥𝑧 times two 𝑥 squared plus 𝑧 squared plus two
𝑥𝑧.
We then check whether the resulting polynomials within each set of parentheses can be
factored. In this case, both polynomials are prime. Therefore, we have that four 𝑦 squared times two 𝑥 squared minus two 𝑥𝑧 plus 𝑧
squared times two 𝑥 squared plus two 𝑥𝑧 plus 𝑧 squared represents the full
factorization of 16𝑥 to the fourth power 𝑦 squared plus four 𝑦 squared 𝑧 to the
fourth power.
