Too many monics

Algebra Level 5

Monic quadratic polynomials \(P(x)\) and \(Q(x)\) have the property that \(P(Q(x))\) has zeros at \(x = -23, -21, -17,\) and \(-15\), and \(Q(P(x))\) has zeros at \(x = -59, -57, -51,\) and \(-49\). What is the sum of the minimum values of \(P(x)\)and \(Q(x)\) ?

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