Final answer:
The Remainder Theorem is used to determine if (x - 6) is a factor of the polynomial by substituting x = 6 into the polynomial p(x). The remainder is zero, so (x - 6) is a factor of p(x). The correct answer is a) 0.
Step-by-step explanation:
The question asks whether (x - 6) is a factor of the polynomial p(x) = 9x^3 - 54x^2 + 6x - 27. To determine if (x - 6) is a factor, we check if the remainder is zero when we divide p(x) by (x - 6). This process is known as the Remainder Theorem, which states that if you substitute the value of x that makes (x - 6) zero into the polynomial and the result is also zero, then (x - 6) is indeed a factor of the polynomial.
Let's apply the Remainder Theorem by substituting x = 6 into p(x):
p(6) = 9(6)^3 - 54(6)^2 + 6(6) - 27
p(6) = 9(216) - 54(36) + 36 - 27
p(6) = 1944 - 1944 + 36 - 27
p(6) = 0
Since the remainder is zero, (x - 6) is indeed a factor of the polynomial p(x).
The answer to the student's question is a) 0.