Let's start by solving the polynomial equation [tex]\( x^6 - 16x^2 = 4x^4 - 64 \)[/tex].
First, let's rearrange the terms of the equation:
[tex]\[ x^6 - 16x^2 - 4x^4 + 64 = 0 \][/tex]
Next, we combine like terms and rewrite the polynomial:
[tex]\[ x^6 - 4x^4 - 16x^2 + 64 = 0 \][/tex]
Given that [tex]\( \pm 2i \)[/tex] are already known complex roots of the equation, we need to find the remaining roots.
We can factorize the polynomial equation to find its roots. We already know that:
[tex]\[ x^2 = -4 \][/tex]
[tex]\( x = \pm 2i \)[/tex]
Now let’s focus on finding the real roots. We solve the polynomial to see that the real roots are obtained, simplifying further:
[tex]\[ x^6 - 4x^4 - 16x^2 + 64 = 0 \][/tex]
By solving the equation with known roots [tex]\( \pm 2i \)[/tex], we can rewrite the polynomial in a more factorized form, solving for the remaining real roots.
With the information provided:
- The roots of the polynomial include both real and complex parts.
- The known complex roots [tex]\( \pm 2i \)[/tex].
We simplify the polynomial to yield the following real roots:
[tex]\[ \{-2, 2\} \][/tex]
Therefore, the complete set of roots for the polynomial equation [tex]\( x^6 - 16x^2 - 4x^4 + 64 \)[/tex] are:
[tex]\[ -2, 2, -2i, 2i \][/tex]
Based on the solution, the other roots apart from the complex roots [tex]\(\pm 2i\)[/tex], the real roots are:
[tex]\[ \boxed{-2 \text{ and } 2} \][/tex]
