If you factorize the polynomial in two steps, you will see it more clearly.
easily factorizes into two factors:
Each of the above factors factorizes into two others by virtue of the identity
The factor (x²-2) does not have a rational factor, but it can still be factored as
and the second one, as you have already done:
So the polynomial actually has four factors, two of which are irrational, and four corresponds to the degree of the polynomial.
it should be quite clear why the four zeroes are where they are.
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