Final answer:
The statement that if 'a' is a zero of multiplicity 'm' of the polynomial 'p(x)', then 'a' must be a factor of 'p(x)' upon complete factorization, is correct. This aligns with the Factor Theorem which confirms that (x - a) would be a factor of 'p(x)' appearing 'm' times.
Step-by-step explanation:
If a is a zero of multiplicity m of the polynomial p(x), then the statement that a must be a factor of p(x) when we factor p(x) completely is Correct. This is based on one of the fundamental theorems in algebra, specifically the Factor Theorem, which states that if a number a is a root (or zero) of the polynomial p(x), then (x - a) is a factor of that polynomial. The multiplicity m indicates how many times the factor (x - a) will appear in the factored form of p(x). Consequently, if a polynomial has a zero of multiplicity m, the factor (x - a) will be raised to the power of m in the polynomial's factored form. To illustrate, for a polynomial p(x) = (x - a)^m * q(x), where q(x) is a polynomial without a as a root, the factor (x - a)^m shows the presence of a as a zero with multiplicity m. Hence, the factorization of p(x) distinctly includes the factor (x - a)^m, confirming the correctness of the statement.