There are no common zeroes within the system:
P21 has no rational factors; P22 factorises x(2x-3), so its two zeroes are 0 and 3/2; x-1 is linear with zero 1.
There is no common intersection, although P21 and P22, P21 and P23, and P22 and P23 intersect:
P21 and P22: x2+2x-1=2x2-3x, x2-5x+1=0 has two intersection points (at (5+√21)/2 and (5-√21)/2);
P21 and P23: x2+2x-1=x-1, x2+x=0 has two intersection points (at 0 and -1);
P22 and P23: 2x2-3x=x-1, 2x2-4x+1=0 has two intersection points (at 1+√2/2 and 1-√2/2).
Conclusion: the polynomials are independent.