Given any polynomial p(x) and any number a, which polynomial is always a factor of p(x) − p(a)?

1. The unit's first-degree-factor lemma proves that x − a always divides p(x) − p(a). 2. The proof writes p(x) − p(a) = sum of l(x² − a²) + m(x − a) terms, each divisible by x − a. 3. This works for any polynomial degree. 4. Combined with p(a) = 0, this gives the full factor theorem. 5. So x − a is always a factor of p(x) − p(a). _Source: SCERT Kerala Std X Mathematics Part-2, Chapter 9-11 (pp 227-246, 2019 ed.)._