What is the completely factored form of the polynomial x^5 - x^3 - 8x^2 + 8?

To determine why the completely factored form of the polynomial ( x^5 - x^3 - 8x^2 + 8 ) is represented accurately in the choice provided, we need to break down the process of factoring the polynomial.

First, it's helpful to notice that the polynomial can be grouped or rearranged to facilitate factoring. By examining the original polynomial, we look for common factors or roots. Testing simple values like ( x = 2 ) can be useful; substituting ( 2 ) into the polynomial yields zero, indicating that ( (x - 2) ) is a factor.

Next, we can divide the original polynomial by ( (x - 2) ) to find the remaining factors. After performing polynomial long division, we obtain a polynomial that simplifies to ( x^4 + x^3 - 6x^2 - 6x + 4 ).

Continuing the factorization of this quartic polynomial reveals that it can be factored further into two quadratic factors. Further manipulation shows that it can be factored as ( (x^2 + 2x + 4)(x^2 - 2x + 2) ). The ( x^2