Question

X⁸-1=0(x⁴-1)*(x⁴+1)=0​

Answer 1

The equation you provided, X⁸-1=0, can be factored as follows:

X⁸ - 1 = (X⁴ - 1)(X⁴ + 1) = 0

Now, let's solve each factor separately:

1. X⁴ - 1 = 0: This is a difference of squares, so we can further factor it as: (X² - 1)(X² + 1) = 0

   Factoring once more: (X - 1)(X + 1)(X² + 1) = 0

   Therefore, we have three solutions for this factor: X = 1, X = -1, and X² + 1 = 0.

2. X⁴ + 1 = 0: This equation doesn't factor further over the real numbers.

In summary, the solutions to the equation X⁸ - 1 = 0 are X = 1, X = -1, and the complex solutions that satisfy X² + 1 = 0.