Question Video: Factorizing by Completing the Square Mathematics

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.