X^3/2 - 2x^3/4 + 1 = 0

solve the equation (hint: it is quadratic in form)

The correct factorization of this polynomial is (x^3/4 - 1) (x^3/4 - 1)

The reason this can be factored is that the middle term has half of the exponent as the lead term. So we can just substitute them to a more standard from and then substitute back at the end. So, for ease sake, we'll make the lead exponent 2 and the middle exponent 1.

x^2 - 2x + 1

Now to factor this, we look for factors of 1 that multiply to - 2. This is clearly - 1 and - 1. So we use them in a parenthesis with the x term.

(x - 1) (x - 1)

Now, since we put x in for x^3/4 at the beginning, we make that switch back now.

(x^3/4 - 1) (x^3/4 - 1)