Video Transcript
Factorize fully eight 𝑦 to the fourth power 𝑛 squared plus 162𝑧 to the fourth
power 𝑛 squared.
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 two 𝑛 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 𝑦 to the fourth power plus 81𝑧 to the fourth power.
We notice that the polynomial in the parentheses contains two perfect square
terms. So, we will attempt to factor this expression by completing the square. To use this method, we will need to manipulate the expression in the parentheses 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 𝑦 to the fourth power and 𝑏
squared to be 81𝑧 to the fourth power, then our value of 𝑎 is the square root of
𝑎 squared, which is equal to two 𝑦 squared. And our value of 𝑏 is the square root of 𝑏 squared, which is equal to nine 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 𝑦 squared times nine 𝑧 squared, which is 36𝑦
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 36𝑦 squared 𝑧
squared minus 36𝑦 squared 𝑧 squared. Our expression with these new terms is two 𝑛 squared times four 𝑦 to the fourth
power plus 36𝑦 squared 𝑧 squared plus 81𝑧 to the fourth power minus 36𝑦 squared
𝑧 squared.
We can now factor the first three terms in the parentheses as a perfect square
trinomial, giving us two 𝑦 squared plus nine 𝑧 squared squared. Now we have a difference of squares, since the expression within the parentheses is
being squared and 36𝑦 squared 𝑧 squared is a perfect square, specifically, the
square of six 𝑦𝑧, 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 𝑦 squared
plus nine 𝑧 squared minus six 𝑦𝑧 times two 𝑦 squared plus nine 𝑧 squared plus
six 𝑦𝑧.
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 two 𝑛 squared times two 𝑦 squared minus six 𝑦𝑧 plus nine 𝑧
squared times two 𝑦 squared plus six 𝑦𝑧 plus nine 𝑧 squared represents the full
factorization of eight 𝑦 to the fourth power 𝑛 squared plus 162𝑧 to the fourth
power 𝑛 squared.
