Video Transcript
Factorize fully four 𝑥 to the fourth power plus nine plus eight 𝑥 squared.
We notice that the polynomial contains two perfect square terms: four 𝑥 to the
fourth power and nine. 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 four 𝑥 to the fourth power and 𝑏
squared to be nine, 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 three. Then, our middle term is equal to two 𝑎𝑏, or sometimes negative two 𝑎𝑏. Two 𝑎𝑏 comes out to two times two 𝑥 squared times three, which is 12𝑥
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 12𝑥 squared minus
12𝑥 squared. Our expression with these new terms is four 𝑥 to the fourth power plus 12𝑥 squared
plus nine plus eight 𝑥 squared minus 12𝑥 squared.
We can now factor the first three terms as a perfect square trinomial, giving us two
𝑥 squared plus three squared. Then we can combine the like terms, giving us negative four 𝑥 squared. Now we have a difference of squares, since the expression within the parentheses is
being squared and four 𝑥 squared is a perfect square, specifically the square of
two 𝑥, 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 three minus two 𝑥 times two 𝑥 squared plus three plus two 𝑥. We then 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 𝑥 squared minus two 𝑥 plus three times two 𝑥 squared plus two
𝑥 plus three represents the full factorization of four 𝑥 to the fourth power plus
nine plus eight 𝑥 squared.
