Final answer:
The statement C, which indicates that the exponents on each term do not follow the pattern for a binomial expansion, is correct. The polynomial x³ – 4x² + 16x – 64 does not match the form required by the binomial theorem, as the exponents don't decrease uniformly, and the coefficients do not match those generated by Pascal's triangle.
Step-by-step explanation:
The statement that explains why the polynomial x³ – 4x² + 16x – 64 cannot be factored using the binomial theorem is C. The exponents on each term do not follow the pattern for a binomial expansion. The binomial theorem allows us to expand expressions that are raised to an integer power and can be written in the form (a + b)n. It yields a sum of terms of the form binomial coefficients times powers of the two terms, where the sum of the exponents in each term is equal to n. For a polynomial to be considered the result of a binomial expansion, it must follow the general form of the theorem, with a uniform decrease in exponents from term to term and binomial coefficients that are consistent with Pascal's triangle.
Option A, which mentions the signs not following the right pattern, is relevant for other forms of factoring, like factoring by grouping or when looking for patterns in signs associated with factoring perfect squares or cubes but is not directly related to the binomial theorem. Option B, which mentions the first and last terms not being perfect cubes, would be relevant if we were looking to factor the expression as a difference of cubes, not for binomial expansion.
To apply the binomial expansion, we look for terms that could be represented as a combination of an and bn terms whose exponents add up to n, but in our case with the polynomial x³ – 4x² + 16x – 64, we find this is not the case. Thus, it cannot be factored by applying the binomial theorem because it does not fit into the form required by the theorem which is established by its series expansion representation.