Final answer:
The factors of the polynomial 9k^2 - 42k + 49 are (3k - 7)(3k - 7) because it is a perfect square trinomial and the middle term indicates that both binomial factors must have the same sign.
Step-by-step explanation:
The factors of the polynomial 9k^2 - 42k + 49 are (3k - 7)(3k - 7) because this polynomial is a perfect square trinomial. A perfect square trinomial is formed when a binomial is squared, which results in the first term being a perfect square, the last term being a perfect square, and the middle term being twice the product of the binomial's terms.
The constant term, 49, is the square of 7 and the coefficient of the middle term, 42, is indeed twice the product of 3k and 7 (2*3k*7 = 42), which confirms the factors are both (3k - 7). Option (3k + 7)(3k + 7) would yield a positive middle term, and (3k + 7)(3k - 7) would yield a different constant term, neither matching the original polynomial.
Thus, the correct answer is B) The polynomial is a perfect square trinomial, although option C) is also correct, it does not fully explain why the factors are (3k - 7) twice. Option A) is not entirely correct because it does not consider the square nature of the polynomial.