Question

directions: factor each polynomial

1. $x^4 - 36$

Explanation:

Step1: Recognize as difference of squares

The polynomial \(x^4 - 36\) can be written as \((x^2)^2 - 6^2\), which is a difference of squares \(a^2 - b^2=(a + b)(a - b)\) where \(a = x^2\) and \(b = 6\).
\(x^4 - 36=(x^2 + 6)(x^2 - 6)\)

Step2: Check if further factoring is possible

Now, \(x^2 - 6\) is also a difference of squares (\(x^2-(\sqrt{6})^2\)), so we can factor it further using the same formula. \(x^2 - 6=(x+\sqrt{6})(x - \sqrt{6})\)

Answer:

\((x^2 + 6)(x+\sqrt{6})(x - \sqrt{6})\)