If x^{n}-py^{n}+qz^{n} is divisible by x^{2}+abyz-bzx-axy, then what is \frac{p}{a^{n}}-\frac{q}{b^{n}} equal to?

Explanation

The divisor factors as (x-ay)(x-bz). For divisibility, x=ay and x=bz must be roots of the dividend. Substituting these gives a^ny^n - py^n + qz^n = 0 \implies qz^n = (p-a^n)y^n and b^nz^n - py^n + qz^n = 0 \implies py^n = (b^n+q)z^n. Multiplying them yields pq = (p-a^n)(b^n+q). Expanding and simplifying gives pb^n - qa^n = a^nb^n. Dividing by a^nb^n leaves \frac{p}{a^n} - \frac{q}{b^n} = 1.

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