One of the factors of p3x + p2(x - y) - p(y + z) - z is :

Concept: To factorise a multi-term algebraic expression, group the terms so that each group shares a common monomial factor, then check whether the resulting groups share a common binomial factor. Pulling out that shared binomial gives one valid factor of the whole expression.

Application:

1. Expand p3x + p2(x - y) - p(y + z) - z by distributing p2 over (x - y) and -p over (y + z): p3x + p2x - p2y - py - pz - z.

2. Group the six terms by shared monomial factors: (p3x + p2x) - (p2y + py) - (pz + z).

3. Factor each group separately: p2x(p + 1) - py(p + 1) - z(p + 1).

4. All three groups now carry the common binomial factor (p + 1). Extracting it gives (p + 1)(p2x - py - z).

5. So the original expression equals (p + 1)(p2x - py - z), which means p2x - py - z is one of its factors.

Cross-check: Expand (p + 1)(p2x - py - z) back out: p3x - p2y - pz + p2x - py - z, which regroups to p3x + p2x - p2y - py - pz - z — exactly the expanded original expression, confirming the factorisation.