To find the zeros of a polynomial you have to find where the graph would intersect the x axis. To do this, you have to find some way to either graph the equation, or as is better here, to factor it.
Factoring the polynomial x4 - 2x2 + 1, you come up with:
(x2 -1)(x2 -1)
They are both the same so you only have to set one equal to zero to find the x intercept(s).
x2 - 1 = 0
x2 = 1
x = √1
x = ± 1
That means that there are 2 x intercepts, at 1 and -1, but you have 2 factors that are exactly the same, so you have a multiplicity of zeros. So 1 has a multiplicity of 2, and -1 has a multiplicity of 2.