Video Transcript
Factorize fully 𝑧 to the fourth power plus 2,500𝑥 to the fourth power.
We notice that the polynomial contains two perfect square terms. So, we will attempt to factor this expression by completing the square. To use this method, we need to manipulate the expression 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 𝑧 to the fourth power and 𝑏 squared to
be 2,500𝑥 to the fourth power, then our value of 𝑎 is the square root of 𝑎
squared, which is equal to 𝑧 squared. And our value of 𝑏 is the square root of 𝑏 squared, which is equal to 50𝑥
squared. Then, our middle term is equal to two 𝑎𝑏, or in some cases negative two 𝑎𝑏. Two 𝑎𝑏 comes out to two times 𝑧 squared times 50𝑥 squared, which is 100𝑧 squared
𝑥 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 100𝑧 squared 𝑥
squared minus 100𝑧 squared 𝑥 squared. Our expression with these new terms is 𝑧 to the fourth power plus 100𝑧 squared 𝑥
squared plus 2,500𝑥 to the fourth power minus 100𝑧 squared 𝑥 squared.
We can now factor the first three terms as a perfect square trinomial, giving us 𝑧
squared plus 50𝑥 squared squared. Now, we have a difference of squares since the expression within the parentheses is
being squared and 100𝑧 squared 𝑥 squared is a perfect square, specifically the
square of 10𝑧𝑥, 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 𝑧 squared plus
50𝑥 squared minus 10𝑧𝑥 times 𝑧 squared plus 50𝑥 squared plus 10𝑧𝑥.
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 𝑧 squared minus 10𝑧𝑥 plus 50𝑥 squared times 𝑧 squared plus
10𝑧𝑥 plus 50𝑥 squared represents the full factorization of 𝑧 to the fourth power
plus 2,500𝑥 to the fourth power.
