alleryGiven any polynomial p(x) and any number a, which polynomial is always a factor of p(x) − p(a)?
Ax + a
Bp(x) − a
Cx − a
Dp(a) − a
Answer & Solution
Correct answer: C. x − 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.)._
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